4.4: Exercise Answers
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Exercise 1
1. Statement
2. Statement
3. Not a statement (question)
4. Statement
5. Not a statement (command)
6. Not a statement (command/request)
7. Statement
8. Statement
9. Statement
10.Statement
11. Not a statement (question)
12. Not a statement (exclamation)
13. Not a statement (command)
14. Statement
15. Statement
Exercise 2
1. Argument. Conclusion: The woman in the hat is not a witch.
2. Not an argument
3. Argument. Conclusion: Albert won’t be willing to help me wash the dishes.
4. Not an argument
5. Not an argument
6. Not an argument
7. Not an argument
8. Argument. Conclusion: Obesity has become a problem in the U.S.
9. Not an argument
10. Argument. Conclusion: Albert isn’t a fireman.
11. Argument. Conclusion: Charlie and Violet don’t sweat.
12. Argument (explanation). Conclusion: I forgot to lock the door.
13. Not an argument
14. Argument (explanation). Conclusion: Apple stole some of Samsung’s patents.
15. Argument. Conclusion: No one who gets frostbite while on K2 will ever survive.
Exercise 3
1. Explanation. Conclusion: Wanda rode the bus today.
2. Explanation. Conclusion: Wanda has not picked up her car from the shop.
3. Argument. Conclusion: Bob rode the bus to work today.
4. Argument. Conclusion: It can’t be snowing right now.
5. Explanation. Conclusion: Some people with schizophrenia hear voices in their head.
6. Argument. Conclusion: Fracking should be allowed.
7. Argument. Conclusion: Wanda did not ride the bus today.
8. Argument. Conclusion: The Tigers will not win their game against the Pirates.
9. Argument. Conclusion: No one living in Pompeii could have escaped
before the lava from Mt. Vesuvius hit.
10. Explanation: When a person moves to Cincinnati, their allergies worsen.
Exercise 4
1
.
1. There is nothing wrong with consensual sexual and economic relations between adults.
2.
3. Therefore, there is nothing wrong with prostitution. (from 1, 2 independently)
2
.
1. Multiple surveys done with prostitutes show that a high percentage of them report having been sexually abused as children.
2. Therefore, prostitution involves women who were typically abused as children. (from 1)
3. Therefore, prostitution is wrong. (from 2)
3
.
1. There was warm water in the cabin’s tea kettle.
2. There was wood still smoldering in the fireplace.
3. Therefore, someone was in this cabin recently. (from 1-2)
4. Tim has been with me the whole time.
5. Therefore, the person in the cabin couldn’t have been Tim. (from 3-4)
6. Therefore, there is someone else in these woods. (from 6)
4
.
1. Marla Runyan ran in the Sydney 2000 Olympics when she was blind.
2. Therefore, it is possible to be blind and yet run in the Olympics. (from1)
5
.
1. The bridge was out.
2. Therefore, the train had to take a longer, alternate route. (from 1)
3. Therefore, the train was late. (from 2)
6
.
1. Iran has threatened to destroy Israel multiple times.
2. If Iran had a nuclear missile, it could destroy Israel.
3. Therefore, if Iran had a nuclear missile, Israel is not safe. (from 1-2)
4. Iran has been developing enriched uranium.
5. Enriched uranium is the most difficult part of the process of building a nuclear weapon—every other part of the process is easy compared to that.
6. Therefore, Israel is not safe. (from 3-5)
7
.
1. All professional hockey players are missing front teeth.
2. Martin is a professional hockey player.
3. Therefore, Martin is missing his front teeth. (from 1-2)
4. Almost all professional athletes who are missing their front teeth have false teeth.
5. Therefore, Martin probably has false teeth. (from 3-4)
8
.
1. Every time I have eaten crab Rangoon at China Food restaurant, I have had stomach troubles afterward.
2. Therefore, probably anyone who eats crab Rangoon at China Food restaurant will have stomach troubles afterward. (from 1)
3. Bob ate the crab Rangoon at China Food restaurant.
4. Therefore, Bob will probably have stomach troubles afterward. (from 2-3)
9
.
1. Albert and Caroline like to run together.
2. Albert never runs without Caroline.
3. Therefore, any time Albert is running, so is Caroline. (from 1-2)
4. Albert is panting hard.
5. Therefore, Albert looks like he has just run (from 4)
6. Therefore, Caroline has probably just ran as well. (from 3, 5)
10
.
1. Jeremy’s prints would be expected to be on his own gun.
2.
3. Therefore, just because Jeremy’s prints were on the gun that killed Tim and the gun was registered to Tim, it doesn’t follow that Jeremy killed Tim. (from 1-2)
Exercise 5
(Note: there are many possible counterexamples to the arguments that are invalid; you don’t have to have the counterexamples I provide to be correct.)
1. Invalid.
2. Invalid. Counterexample: suppose Bob just became a fireman, so he has never actually put out any fires.
3. Invalid. Counterexample: Although Gerald probably knows how to teach mathematics, suppose he has just had a traumatic brain injury and no longer knows how to teach mathematics. And suppose the injury is recent enough for him not to have lost his job as a mathematics professor yet.
4. Invalid. Counterexample: A similar counterexample as #3 would work equally as well here.
5. Valid.
6. Invalid.
7. Invalid. Counterexample: suppose that although Orel Hershizer believes that God exists, in fact God doesn’t exist. In that case, Orel can’t communicate with God, since you can’t communicate with something that doesn’t exist (i.e., communication is a two-way interaction).
8. Valid.
9. Valid.
10.Valid.
Exercise 6
(Note: there is more than one possible correct answer to some of these.)
1. Anyone who rides horses is a cowboy.
2. Driving over the speed limit is wrong.
3. It is raining.
4. Olaf is an elf.
5. Any time a person has a choice of who to take to homecoming, they will take the person they like the most.
6. I have looked at the watch in frequent intervals—much more often than every 12 hours.
7. Only those who have drank too much fall out of apartment windows.
8. Mark is on Earth and is unassisted by any devices that help him overcome the Earth’s gravity.
9. Any nation in which there is a large discrepancy between net worths of different races is a racist nation.
10. The water is at sea level.
11. First missing premise: We should not allows policies that have the potential of taking innocent lives, unless there is a very good reason to do so. Second missing premise: there is no very good reason to allow capital punishment.
12. First missing premise: We should not allow policies that take working class jobs away from working class folks, unless there is some very good reason to do so. Second missing premise: there is no very good reason to allow immigration that would offset the harm done to working class folks.
13. Any fair economic exchange between consenting adults should be allowed.
14. Anything that privileges using a student-athlete to make money for the college over that student athlete’s education should be banned.
15. Any student who receives an F in a course should have studied more for that course.
Exercise 7
1. Discounting
2. Assuring
3. Discounting
4. Assuring
5. Discounting
6. Guarding
7. Assuring
8. Discounting
9. Discounting
10. Assuring
Exercise 8
1. Not truth functional.
2. Truth functional: Tom is a fireman. Tom is a father.
3. Not truth functional.
4. Not truth functional.
5. Truth functional: Cameron Dias has had several relationships. Cameron Dias has never married.
6. Not truth functional.
7. Not truth functional.
8. Not truth functional.
9. Truth functional: Jack is a cowboy. Jill is a cowboy.
10.Truth functional: Josiah is Amish. Josiah is a drug dealer.
11.Truth functional: The Tigers are the best baseball team in the state. The Tigers are not as good as the Yankees.
12.Truth functional: Bob went to the beach to enjoy some rest. Bob went to the beach to enjoy some relaxation.
13.Truth functional: Lauren isn’t the fastest runner on the team. Lauren is fast enough to have made it to the national championship.
14.Truth functional: The ring is beautiful. The ring is expensive.
15.Truth functional: It is sad that many Americans to not know where their next meal will come from. It is true that many Americans do not know where their next meal will come from.
Exercise 9
1. B v T
2. ~S
3. ~B
4.  C Ŋ D
9. ~B
10.~A
Exercise 10
1. (~A Ŋ ~S) Ŋ M (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
2. ~S Ŋ ~H (The main operator is the conjunction.)
3. P Ŋ ~N (The main operator is the conjunction.)
4. (~S Ŋ ~F) Ŋ A (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
5. ~C Ŋ (F Ŋ B) (The main operator is the first dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
6. (T Ŋ ~S) Ŋ L (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
7. E Ŋ W (There is only one truth functional operator, the conjunction. So that is by default the main operator!)
8. (T v L) Ŋ ~S (The main operator is the conjunction.)
9. (B v F) Ŋ ~(P v D) (The main operator is the conjunction.)
10. (A Ŋ J) Ŋ E (The main operator is the second dot—in this case it doesn’t actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator.)
11. (A Ŋ J) v (C Ŋ M) (The wedge is the main operator.)
12. S Ŋ ~H (The main operator is the conjunction.)
Exercise 11
1
.
a. P = Coral is a plant; A = Coral is an animal
b. ~(P Ŋ A)
c. Main operator is the negation
2
.
a. A = Protozoa are eukaryotes; B = Chimpanzees are eukaryotes; C
= Protozoa are animals; D = Chimpanzees are animals
b. (A Ŋ B) Ŋ ~(C Ŋ D)
c. Main operator is the second dot
3
.
a. C = Chimpanzees are prokaryotes; P = Protozoa are prokaryotes
b. ~(C v P)
c. Main operator is the negation
4
.
a. C = China has not signed the Kyoto Protocol; U = The United
States has not signed the Kyoto Protocol
b. ~(C v U)
c. Main operator is the negation
5
.
a. C = Chevrolet will support the Olympic team; M = McDonald’s will
support the Olympic team
b. (C v M) Ŋ ~(C Ŋ M)
c. Main operator is the first dot
6
.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. L v M
c. Main (and only) operator is the wedge
7
.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. ~(L v M)
c. Main operator is the negation
8
.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. L Ŋ M
c. Main (and only) operator is the dot
9
.
a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad
memory
b. ~(L Ŋ M)
c. Main operator is the negation
10
.
a. C = Chevrolet will support the Olympic team this year; M =
McDonald’s will support the Olympic team this year
b. ~(C v M)
c. Main operator is the negation
11
.
a. S = Mother Theresa is a saint; C = Mother Theresa has been
canonized by the Catholic Church
b. S Ŋ ~C
c. Main operator is the dot
12
.
a. T = Paul Tergat was the best distance runner of the last two
decades; G = Haile Gebrselassie was the best distance runner of
the last two decades; R = Jim Ryun was the best distance runner of
the last two decades
b. (T v G) Ŋ ~R
c. Main operator is the dot
13
.
a. R = Jim Ryun was the best high school miler of all time; W = Jim
Ryun ran a slower time than Alan Webb
b. R Ŋ W
c. Main (and only) operator is the dot
14
.
a. A = Paul Tergat knows how to play hockey; B = Haile Gebrselassie
knows how to play hockey; C = Paul Tergat knows how to play
soccer; D = Haile Gebrselassie knows how to play soccer
b. ~(A v B) Ŋ (C Ŋ D)
c. Main operator is the first dot
15
.
a. B = Ethiopians are good bobsledders; T = Ethiopians are good
tennis players; D = Ethiopians are good distance runners
b. ~(B v T) Ŋ D
c. Main operator is the dot
16
.
a. S = Before Helen Keller met Annie Sullivan she could speak; R =
Before Helen Keller met Annie Sullivan she could read; C = Before
Helen Keller met Annie Sullivan she could communicate
b. (~S Ŋ ~R) Ŋ ~C
c. Main operator is the second dot
17
.
a. C = Helen Keller learned to communicate; S = Helen Keller learned
to play soccer; B = Helen Keller learned to play baseball
b. C Ŋ (~S Ŋ ~B)
c. Main operator is the first dot
18
.
a. F = Tom is allowed to play football; S = Tom is allowed to play
soccer
b. (F v S) Ŋ ~(F Ŋ S)
c. Main operator is the first dot
19
.
a. E = Tom will major in engineering; P = Tom will major in physics; B
= Tom will major in business; S = Tom will major in sociology
b. (E Ŋ P) v (B Ŋ S)
c. Main operator is the wedge
20
.
a. X = Cartman is xenophobic; R = Cartman is a racist; M = Cartman
is a murderer; T = Cartman is the thief
b. (X Ŋ R) Ŋ (~M Ŋ ~T)
c. Main operator is the second dot
Exercise 12
1. Invalid
2. Valid
3. Invalid
4. Valid
5. Invalid
6. Invalid
7. Valid
8. Valid
9. Valid
Exercise 13
1. T ⊃ I (T = The Tigers will win; I = The Indians will lose their star pitcher)
2. H ⊃ P (P = Tom will pass the class; H = Tom does all of his homework)
3. R ⊃ G (R = The car will run; G = The car has gas)
4. A ⊃ C (A = You are asking me about your grade; C = You care about your grade)
5. F ⋅ (T ⊃ B) (F = Frog will swim without his bathing suit; T = Toad will swim; B = Toad is wearing a bathing suit)
6. ~O ⊃ M (O = Obama is not a U.S. citizen; M = I am a monkey’s uncle)
7. T ⊃ ~F (T = Toad wears his bathing suit; F = Toad wants Frog to see him in his bathing suit)
8. ~P ⊃ (S v L) (P = Tom passes his exam; S = Tom is stupid; L = Tom is lazy)
9. H ⊃ W (W = Bekele will win the race; H = Bekele stays healthy)
10. (S v I) ⊃ ~W (S = Bekele is sick; I = Bekele is injured; W = Bekele will win the race)
11. P ⊃ (C ⋅ ~S) (P = Bob will become president; C = Bob runs a good campaign; S = Bob says something stupid)
12. T ⊃ P (T = That plant has three leaves; P = That plant is poisonous)
13. P ⊃ T (T = That plant has three leaves; P = That plant is poisonous)
14. P ⊃ T (T = That plant has three leaves; P = That plant is poisonous)
15. P ⊃ T (T = That plant has three leaves; P = That plant is poisonous)
16. N ⊃ O (O = Olga will swim in the open water; N = There is a shark net present)
17. O ⊃ N (O = Olga will swim in the open water; N = There is a shark net present)
18. O ⊃ B (O = Olga is swimming; B = Olga is wearing a bathing suit)
19. N ⊃ ~B (N = Olga is in Nice; B = Olga wears a bathing suit)
20. T ⊃ B (T = Terrence pulls Philip’s finger; B = Something bad will happen)
Exercise 14
1. Equivalent
2. Equivalent
3. Equivalent
4. Equivalent
5. Not equivalent
6. Equivalent
7. Not equivalent
8. Equivalent
9. Not equivalent
10. Not equivalent
Exercise 15
1. Contingent
2. Tautology
3. Tautology
4. Contradiction
5. Tautology
6. Contingent
7. Contingent
8. Tautology
9. Contingent
10. Tautology
Exercise 16
1.
Simplification 1
Simplification 1
Addition 2
Addition 3
Conjunction 4, 5
2.
Disjunctive syllogism 2, 3
Modus ponens 1, 4
Modus tollens 2, 5
3.
Simplification 3
Modus tollens 1, 4
Disjunctive syllogism 2, 5
4.
Simplification 2
Hypothetical syllogism 1, 3
Modus ponens 4, 5
Simplification 2
Conjunction 6, 7
5.
Hypothetical syllogism 3, 4
Addition 1
Constructive dilemma 2, 5, 6
6.
Simplification 2
Modus tollens 1, 4
Disjunctive syllogism 3, 5
7.
Simplification 2
Modus ponens 3, 4
Simplification 2
Disjunctive syllogism 5, 6
Conjunction 4, 7
Modus ponens 1, 8
8.
Simplification 2
Disjunctive syllogism 3, 5
Simplification 2
Modus tollens 1, 7
Disjunctive syllogism 4, 8
Modus ponens 6, 9
Modus tollens 8, 10
Addition 10
Exercise 17
1
.
1. A ⋅ B
2. (A v C) ⊃ D /∴ A ⋅ D
3. A Simplification 1
4. A v C Addition 3
5. D Modus ponens 2, 4
6. A ⋅ D Conjunction 3, 5
2
.
1. A
2. B /∴ (A v C) ⋅ B
3. A v C Addition 1
4. (A v C) ⋅ B Conjunction 2, 3
3
.
1. D ⊃ E
2. D ⋅ F /∴ E
3. D Simplification 2
4. E Modus ponens 1, 3
4
.
1. J ⊃ K
2. J /∴ K v L
3. K Modus ponens 1, 2
4. K v L Addition 3
5
.
1. A v B
2. ~A ⋅ ~C /∴ B
3. ~A Simplification 2
4. B Disjunctive syllogism 1, 3
6
.
1. A ⊃ B
2. ~B ⋅ ~C /∴ ~A
3. ~B Simplification 2
4. ~A Modus tollens, 1, 3
7
.
1. D ⊃ E
2. (E ⊃ F) ⋅ (F⊃ D) /∴ D ⊃ F
3. E ⊃ F Simplification 2
4. D ⊃ F Hypothetical syllogism 1, 3
8
.
1. (T ⊃ U) ⋅ (T ⊃ V)
2. T /∴ U v V
3. T v U Addition 2
4. U v V Constructive dilemma 1, 2, 3
9
.
1. (E ⋅ F) v (G ⊃ H)
2. I ⊃ G
3. ~(E ⋅ F) /∴ I ⊃ H
4. G ⊃ H Disjunctive syllogism 1, 3
5. I ⊃ H Hypothetical syllogism 2, 4
10
.
1. M ⊃ N
2. O ⊃ P
3. N ⊃ P
4.  M v O Modus ponens 3, 4
6. N v P Constructive dilemma 1, 2, 5
11
.
1. A v (B ⊃ A)
2. ~A ⋅ C /∴ ~B
3. ~A Simplification 2
4. B ⊃ A Disjunctive syllogism 1, 3
5. ~B Modus tollens 3, 4
12
.
1. (D v E) ⊃ (F ⋅ G)
2. D /∴ F
3. D v E Addition 2
4. F ⋅ G Modus ponens 1, 3
5. F Simplification 4
13
.
1. T ⊃ U
2. V v ~U
3. ~V ⋅ ~W /∴ ~T
4. ~V Simplification 3
5. ~U Disjunctive syllogism 2, 4
6. ~T Modus tollens 1, 5
14
.
1. (A v B) ⊃ ~C
2. C v D
3. A /∴ D
4. A v B Addition 3
5. ~C Modus ponens 1, 4
6. D Disjunctive syllogism 2, 5
15
.
1. L v (M ⊃ N)
2. ~L ⊃ (N ⊃ O)
3. ~L /∴ M ⊃ O
4. N ⊃ O Modus ponens 2, 3
5. M ⊃ N Disjunctive syllogism 1, 3
6. M ⊃ O Hypothetical syllogism 4, 5
16
.
1. A ⊃ B
2. A v (C ⋅ D)
3. ~B ⋅ ~E /∴ C
4. ~B Simplification 3
5. ~A Modus tollens 1, 4
6. C ⋅ D Disjunctive syllogism 2, 5
7. C Simplification 6
17
.
1. (F ⊃ G) ⋅ (H ⊃ I)
2. J ⊃ K
3. (F v J) ⋅ (H v L) /∴ G v K
4. F ⊃ G Simplification 1
5. F v J Simplification 3
6. G v K Constructive dilemma 2, 4, 5
18
.
1. (E v F) ⊃ (G ⋅ H)
2. (G v H) ⊃ I
3. E /∴ I
4. E v F Addition 3
5. G ⋅ H Modus ponens 1, 4
6. G Simplification 5
7. G v H Addition 6
8. I Modus ponens 2, 7
19
.
1. (N v O) ⊃ P
2. (P v Q) ⊃ R
3.  N Disjunctive syllogism 3, 4
6. N v O Addition 5
7. P Modus ponens 1, 6
8. P v Q Addition 7
9. R Modus ponens 2, 8
20
.
1. J ⊃ K
2. K v L
3. (L ⋅ ~J) ⊃ (M ⋅ ~J)
4. ~K /∴ M
5. L Disjunctive syllogism 2, 4
6. ~J Modus tollens 1, 4
7. L ⋅ ~J Conjunction 5, 6
8. M ⋅ ~J Modus ponens 3, 7
9. M Simplification 8
Exercise 18
1. All real men are things that wear pink.
2. No dinosaurs are birds.
3. All birds are things that evolved from dinosaurs.
4. Some mammals are not predators. [Already in “Some S are not P”
categorical form.]
5. Some predators are not mammals. [Already in “Some S are not P”
categorical form.]
6. Some things that wander are not things that are lost.
7. No presidents are women.
8. No boxers are rich people.
9. No things that are sleeping are things that are conscious.
10. No things that are conscious are things that are sleeping.
11. All things that end well are things that are well. [Note that this is in a different order than the sentence states. However, if you think about the meaning of the cliché, it should be clear that this is the correct order. What is being said is that as long as something ends well, it is well. It is not saying that the only things that can be well are things that end well.]
12. All things that care are things that are my friends.
13. Some person is a person who loves you.
14. All people are people who are loved by Jesus.
15. All little children are people who are loved by Jesus.
16. Some people are people who don’t love Jesus.
17. All things that use the Appalachian Trail are pedestrians.
18. All presidents are citizens.
19. All Hindus are people who believe in God.
20. No cheap things are good things.
21. Some expensive things are not good things.
22. Some mammals are not things that have legs.
23. Some couples are couples without children.
24. No people are chocolate-haters.
25. Some people are cat-haters.
26. No sharp things are safe things.
27. No poodles are things that can run faster than a cheetah.
28. No professional runner is a person that runs slowly.
29. No baboons are friendly creatures.
30. All things are things that would be eaten by a pig.
Exercise 19
1. Invalid
2. Valid
3. Invalid
4. Invalid
5. Valid
6. Invalid
7. Invalid
8. Invalid
Exercise 20
1. Invalid
2. Invalid
3. Invalid
4. Invalid
Exercise 21
1. Invalid
2. Invalid
3. Invalid
4. Invalid
5. Invalid
6. Valid
7. Invalid
8. Invalid
9. Valid
10. Valid
11. Invalid
12. Invalid
13. Invalid
14. Invalid
15. Invalid
16. Valid
17. Valid
18. Invalid
19. Invalid
20. Invalid
Exercise 22
1. Hasty generalization (you can’t infer something general from just one case here—the sample size is way too small). There is also a sampling bias present: even if many others people from Silverton, CO drove pickups, it doesn’t follow that people generally do. There is a high percentage of trucks in Silverton because the rough roads there almost require trucks.
2. Biased sample: even if he has an adequate sample size, Tom needs to sample from different times during the morning to be sure that he has a representative sample. If morning doves are disproportionately represented during the early morning hours, then his sample will be biased.
3. Even more clearly than the previous example, this one is a biased sample: even if he has an adequate sample size, Tom needs to sample from different times of the day. It is likely that morning doves will be disproportionately represented in the morning, since they are more likely to be out in the morning than other kinds of birds.
4. This example corrects the problems of the previous two: Tom has sampled from different times during the day. As long as he has taken these samples on multiple different days (preferably in different seasons too), then his sample is representative and his generalization is good.
5. Biased sample. Same problem, mutatis mutandis, as #3.
6. This seems to be a good generalization, assuming that he keeps up this regimen on multiple days. The difference, of course, is that instead of making his generalization cover the whole day, his generalization is only about the birds that land in his tree during the night.
7. Biased sample. Of course the home owners will be likely to support a policy that slashes property taxes. Most likely, those on Medicaid (governmental health care support for the elderly) will not be homeowners but will be in nursing home facilities. If the poll had been administered to Medicaid recipients (who are less likely to own homes), the results would likely have been different.
8. This seems a good generalization. Telephone polls are a good way of getting a random sample, and the sample size is large enough if a good random sampling technique is used.
9. Sampling bias because of the biased way the question is asked: “killing innocent children” uses strong, evaluative language and may influence how people answer, making them more likely to choose option b over option a (who wants to say they support “killing innocent children”?).
10. Steve’s problem is that he has gotten a biased sample. Ani Difranco concert-goers are not representative of concert-goers tout court. Since Ani Difranco is very political (and from a feminist perspective), we should expect to see a much higher proportion of such speech at an Ani Difranco concert. In contrast, Tom Petty is about a apolitical as any musician.
11. Biased sample. We should expect students in detention to be less satisfied, on average, than students generally. Thus, since the principal’s survey was only administered to students in detention, the rate of dissatisfaction will be much higher, which will make the sample unrepresentative and the generalization bad.
12. This seems to be a good generalization. Her generalization only covers “all Pistons games” (rather than all NBA games or all professional sports games, more generally) and she has attended many games over many years. Thus the sample seems to be both representative (i.e., non-biased) and large enough.
13. Unlike the last example, Alice’s generalization now applies to all NBA games, but still uses only her experience at Pistons games. But unless we are given some reason for thinking that Pistons games are representative of all NBA games, we should not assume that Pistons games are representative of all NBA games. Thus, the sample is probably biased (although we do not know for sure that it is, we cannot assume it isn’t without further investigation).
14. Even more than the last example, this one is biased sample. Unless we have a good reason for thinking that Pistons games are representative of all professional sporting events, we cannot assume that they are.
15.Although we can understand Bob’s fear, this is clearly a hasty generalization since he is generalization from only one case at one Burger King to all Burger Kings, all the time.
Exercise 23
(Note: for many of these, there is more than one correct answer. The important thing to do is to give the correctly explanation for why the explanation lacks the virtue you have chosen.)
1. This could be any number of them, including: depth (why would the aliens have kidnapped him and then returned him to his home?), power (this explanation cannot be used in a range of different circumstances—a better explanation is simply that he has some kind of amnesia), or simplicity (if we don’t have any other reason to admit there are aliens, then we should simply chalk it up to some kind of amnesia).
2. Modesty. There is no reason she should posit all of those specific details about the badger, even if it was a badger. However, even just saying it’s a badger or a large rodent is an explanation that seems to lack simplicity. If houses naturally creak and windows rattle from the wind, then positing a large rodent seems unnecessary. A better explanation would simply be that the house creaks naturally as it slightly shifts and the wind is rattling the windows.
3. Simplicity and modesty. It is simpler to simply assume that there is someone who looks like Bob, whether or not he is Bob’s identical twin. It is also more modest since positing someone who looks like Bob could include someone that is Bob’s identical twin, but also leaves open the possibility that it’s just an unrelated person who happens to look like Bob. The explanation might also lack power insofar as it raises more questions than it answers. For example, why did Bob never tell you about his identical twin?
4.  Thus, we could also say it lacks power since this kind of explanation doesn’t apply in any other cases we know. A better explanation is that there is someone who looks just like Tom.
5. Modesty. Like #3, a more modest explanation is that this is someone who looks like Tom, whether or not it is Tom’s son. The explanation might also lack depth since we would want to know why you had never seen or heard of Tom’s son for 20 years.
6. The last line is the giveaway: this explanation lacks falsifiability. The reason is that Elise says that there is no way to prove that this happened (she just knows it). The explanation also lacks depth since we would want to know why and how this replacement was done!
7. If this explanation lacks an explanatory virtue, it is probably falsifiability: there is no way (within current science) to show that there wasn’t such a being. Furthermore, it might also lack depth since it raises the question: where did this all-powerful being come from?
8. Modesty. Why think that it is her 5th grade teacher rather than just some person following her? The explanation is far more specific than it needs to be in order to explain the observations she has made. Thus, it lacks modesty.
9. Again, this explanation lacks modesty. Why not just say that it is “an animal” rather than “an escaped zoo animal.” Unless she has some evidence relevant to the escaped zoo animal hypothesis, she should just leave it at the more general “animal” hypothesis. Furthermore, the explanation may be said to lack power, as well. Since most such noises are made by creatures in the wild, not escaped zoo animals, the “creatures in the wild” explanation is more powerful, since it is used to explain a much wider range of similar observations (i.e., hearing rustling in the bushes and sticks cracking on the ground while in the woods).
10. Simplicity. The simpler explanation is that Bill was speeding, not that they had tracked his overdue library book. It also lacks power since most of the time when people are pulled over on the highway it is for speeding, not unreturned library books.
11. This is a good explanation and seems to lack no explanatory virtue.
12. This explanation clearly lacks modesty. Why say that someone was going precisely 13.74 mph over the speed limit rather than saying that they were going over the speed limit (without specifying how far)? That specificity is not justified by the observed facts.
13. Conservativeness. We have no good reason for positing some whole new breed of rats—especially if the claim is that they evolved in her apartment only. This would violate what we know about how evolution works (i.e., we probably need a much larger population for this to happen than the population of rats that are contained in only her apartment). Furthermore, the explanation lacks power since a better explanation that applied to a wider range of circumstances is simply that the rats were not taking the bait.
14. Even more clearly than #13, this one lacks conservativeness. There are no known cases of anything being immortal and this idea violates our understanding of the basic laws of nature. Nothing is immortal.
15. Again, this explanation lacks conservativeness (i.e., it violates our understanding of nature which says that nothing is immortal). A better explanation is that the bullets Bob put in his gun were blanks (cf. the movie, Crash).
Exercise 24
1. Weak: if the painting is hanging in your high school, it probably isn’t a Rembrandt. That is the disanalogy: even if the colors are very similar, almost all Rembrandts hang in galleries, not in high schools.
2. Weak. Although the similarity is that they are both poodles, there is probably some other characteristic that accounted for me being bitten. That is, it probably wasn’t the fact that the dog that bit me was a poodle, but more likely that I was invading its space or it felt threatened, etc. It could have likely been some other breed in the same circumstances. So it isn’t “poodleness” that accounts for the biting. That said, if we had evidence that poodles are much more likely to bite than other breeds then this argument would be stronger.
3. Strong. Unlike, the last one, this argument delivers a much stronger analogy between past events (poodle-encounters and poodle-bitings) and the current event (poodle-encounter).
4. Strong. The relevant similarities are: 1) Van Cleave’s class doesn’t change much from semester to semester, 2) the person has the same abilities as their friend who got the A.
5. Weak. Although both are crimes, there are many relevant differences between committing rape and robbing a bank.
6. Weak. There is no particular relationship between having seats, wheels, and brakes, on the one hand, and being safe to drive, on the other. So having seats, wheels and brakes is not a relevant similarity between the two cars, if what we are interested in is how safe they are.
7. Strong. The car company (Volvo) is a relevant similarity between the old cars and the new car. We can expect similar quality between cars from the same company. In contrast, knowing that a car as wheels, brakes and seats tells us essentially nothing about its quality, including its safety.
8. Strong.
9. Weak. A birthday party and a funeral are not relevantly similar in this case. A funeral is a much more important family event than a birthday party (typically). So we should not expect similarity with respect to a professor’s absence policy when comparing birthday parties to funerals.
10. Weak. Although both may influence happiness, the relevant difference is that whereas heart and brain surgery are typically a matter of life and death (and hence much more likely to be paid for by insurance), cosmetic surgery is not a matter of life and death.
11. Weak. Although a knife and spoon share the property of being eating utensils, that is not a relevant similarity on which we can expect that they will share functional properties like cutting.
12. Whether this famous argument for the existence of God is strong or weak is a matter of some debate. One reason for saying it is a weak argument is that there is a disanalogy between artificial objects and natural objects, since complex natural objects may evolve without being designed by an intelligent designer, whereas no artificial objects (yet) can evolve on their own.
13. Weak. Running the same number of miles as an elite runner is not a relevant similarity for determining how fast one will run a race. The relevant dissimilarity here is that although Bekele runs his mile repeats at close to 4:00 flat, I can only run mine at 5:30. So it is the pace at which one runs, rather than the number of miles one runs, that is the better predictor of how fast one can run a race.
14. Strong. The fact that we are both humans is relevant to determining whether someone will feel pain. Humans all have similar physiology, which is why we should expect that if x causes one person physical pain, then x will also cause anyone else a similar pain. (However, this argument also raises a famous problem in philosophy of mind called “the problem of other minds.” The issue is whether or not we can ever know that people have mental states, such as pain, like my own. Even if you exhibit pain behavior in similar instances in which I experience pain, how do I know that you are actually feeling what I am feeling—that you are having the experience of pain, rather than simply exhibiting pain behavior without have the mental experience of it? Many philosophers have argued that we cannot overcome this problem and must admit that we cannot know whether people other than ourselves actually have mental states like ours.)
15. Again, the common sense answer would be that this is a strong argument based on a strong analogy. Since you and I are both human and share similar perceptual systems, we should expect that we will perceive the world very similarly (even if not exactly the same). (However, we can raise the same “problem of other minds” problem here as I did in #14 above. Suppose we both point at the grass and say that it is green. However, how do I know that your experience of green is like my experience of green? Maybe your experience of green is more like my experience of red and vice versa.)
Exercise 25
1. C is sufficient since any time it is present, the target G is present. Both C and D are necessary, since any time the target G is present, they are present.
2. A and C are sufficient; A and C are also necessary.
3. C and D are necessary. All candidates are sufficient, since there is never a case where a candidate is present but the target is absent. However, the test for sufficient conditions is not rigorous, since there is no case in which the target is absent (hence, it is trivially true that there is no case in which a candidate is present when the target is absent).
4. No one candidate is sufficient (case 1 rules them all out). Only C is necessary.
5. C is sufficient; B, C, and D are necessary.
6. None are sufficient (case 1 rules them all out); A and D are necessary.
7. B is sufficient; A, B and D are necessary.
8. C is sufficient; A, B, C, and D are all necessary. However, the test for necessary conditions is not rigorous, since there is no case in which the target is present (hence, it is trivially true that there is no case in which a candidate is absent when the target is present).
9. B and C are sufficient; C and D are necessary.
10. A and C are sufficient; D is necessary.
Exercise 26
1. Accidental
2. B causes A. Perhaps when we are away from home, Charlie gets less exercise, hence putting on more weight. In that case B is (indirectly) causing A, since our being away from home results in Charlie getting less exercise, which results in him gaining more weight.
3. Common cause. The common cause is simply the factors that make plants grow, such a sunlight, water, and good soil. In this case, A and B are not causing each other, but there is something else (i.e., growth factors of plants) that is causing each one, independently to grow.
4. A causes B. The more bombing, the more stress for the president. And the more stress for the president, the more of his hairs fall out.
5. B causes A. Arguably, the average number of fires per year will influence the number of fire engines needed. Fewer fires would require fewer engines (in which case they’d likely retire some); more fires would require more engines (in which case they’d likely acquire some).
6. Common cause: agrarian societies will have more mules and will also probably pay professors less, since higher education is not as important in an agrarian society.
7. B causes A. Drivers are distracted by the scantily-clad models on the billboards and more distracted drivers causes more accidents.
8. A causes B. The wider one’s waist, the higher the weight due to fat. The higher the weight due to fat, the lower one’s vertical leap.
9. B causes A. The heat causes slower marathon times.
10. Common cause: Ageing. The older one is, the more likely one will have gray hair and the more likely one will have more children or grandchildren. However, neither of these factors are causing the other. Rather, both are being caused independently by a common factor: age.
Exercise 27
1. 1/6
2. 5/6
3. 1/36
4. 1/36
5. 12/36 or 1/3
6. 1/6 + 1/6 – 1/36 = 11/36
7. 2/6
8. 1 – 12/36 = 24/36 = 2/3
9. 1/36
10. 6/36


10.5: Measuring Stellar Masses Part 1
Learning Objectives
By the end of this section, you will be able to:
- Explain why the stars visible to the unaided eye are not typical
- Describe the distribution of stellar masses found close to the Sun
Before we can make our own survey, we need to agree on a unit of distance appropriate to the objects we are studying. The stars are all so far away that kilometers (and even astronomical units) would be very cumbersome to use; so—as discussed in Science and the Universe: A Brief Tour—astronomers use a much larger “measuring stick” called the light-year . A light-year is the distance that light (the fastest signal we know) travels in 1 year. Since light covers an astounding 300,000 kilometers per second, and since there are a lot of seconds in 1 year, a light-year is a very large quantity: 9.5 trillion (9.5 × 10^12) kilometers to be exact. (Bear in mind that the light-year is a unit of distance even though the term year appears in it.) If you drove at the legal US speed limit without stopping for food or rest, you would not arrive at the end of a light-year in space until roughly 12 million years had passed. And the closest star is more than 4 light-years away.
Notice that we have not yet said much about how such enormous distances can be measured. That is a complicated question, to which we will return in Celestial Distances. For now, let us assume that distances have been measured for stars in our cosmic vicinity so that we can proceed with our census.
Small Is Beautiful—Or at Least More Common
When we do a census of people in the United States, we count the inhabitants by neighborhood. We can try the same approach for our stellar census and begin with our own immediate neighborhood. As we shall see, we run into two problems—just as we do with a census of human beings. First, it is hard to be sure we have counted all the inhabitants; second, our local neighborhood may not contain all possible types of people.
Table \(\PageIndex{1}\) shows an estimate of the number of stars of each spectral type 1 in our own local neighborhood—within 21 light-years of the Sun. (The Milky Way Galaxy, in which we live, is about 100,000 light-years in diameter, so this figure really applies to a very local neighborhood, one that contains a tiny fraction of all the billions of stars in the Milky Way.) You can see that there are many more low-luminosity (and hence low mass) stars than high-luminosity ones. Only three of the stars in our local neighborhood (one F type and two A types) are significantly more luminous and more massive than the Sun. This is truly a case where small triumphs over large—at least in terms of numbers. The Sun is more massive than the vast majority of stars in our vicinity.
| Table \(\PageIndex{1}\): Stars within 21 Light-Years of the Sun | |
|---|---|
| Spectral Type | Number of Stars |
| A | 2 |
| F | 1 |
| G | 7 |
| K | 17 |
| M | 94 |
| White dwarfs | 8 |
| Brown dwarfs | 33 |
This table is based on data published through 2015, and it is likely that more faint objects remain to be discovered (see Figure \(\PageIndex{1}\)). Along with the L and T brown dwarfs already observed in our neighborhood, astronomers expect to find perhaps hundreds of additional T dwarfs. Many of these are likely to be even cooler than the coolest currently known T dwarf. The reason the lowest-mass dwarfs are so hard to find is that they put out very little light—ten thousand to a million times less light than the Sun. Only recently has our technology progressed to the point that we can detect these dim, cool objects.
To put all this in perspective, we note that even though the stars counted in the table are our closest neighbors, you can’t just look up at the night sky and see them without a telescope; stars fainter than the Sun cannot be seen with the unaided eye unless they are very nearby. For example, stars with luminosities ranging from 1/100 to 1/10,000 the luminosity of the Sun ( L Sun ) are very common, but a star with a luminosity of 1/100 L Sun would have to be within 5 light-years to be visible to the naked eye—and only three stars (all in one system) are this close to us. The nearest of these three stars, Proxima Centauri, still cannot be seen without a telescope because it has such a low luminosity.
Astronomers are working hard these days to complete the census of our local neighborhood by finding our faintest neighbors. Recent discoveries of nearby stars have relied heavily upon infrared telescopes that are able to find these many cool, low-mass stars. You should expect the number of known stars within 26 light-years of the Sun to keep increasing as more and better surveys are undertaken.
Bright Does Not Necessarily Mean Close
If we confine our census to the local neighborhood, we will miss many of the most interesting kinds of stars. After all, the neighborhood in which you live does not contain all the types of people—distinguished according to age, education, income, race, and so on—that live in the entire country. For example, a few people do live to be over 100 years old, but there may be no such individual within several miles of where you live. In order to sample the full range of the human population, you would have to extend your census to a much larger area. Similarly, some types of stars simply are not found nearby.
A clue that we are missing something in our stellar census comes from the fact that only six of the 20 stars that appear brightest in our sky—Sirius, Vega, Altair, Alpha Centauri, Fomalhaut, and Procyon—are found within 26 light-years of the Sun (Figure \(\PageIndex{2}\)). Why are we missing most of the brightest stars when we take our census of the local neighborhood?
The answer, interestingly enough, is that the stars that appear brightest are not the ones closest to us. The brightest stars look the way they do because they emit a very large amount of energy—so much, in fact, that they do not have to be nearby to look brilliant. You can confirm this by looking at Appendix J, which gives distances for the 20 stars that appear brightest from Earth. The most distant of these stars is more than 1000 light-years from us. In fact, it turns out that most of the stars visible without a telescope are hundreds of light-years away and many times more luminous than the Sun. Among the 9000 stars visible to the unaided eye, only about 50 are intrinsically fainter than the Sun. Note also that several of the stars in Appendix J are spectral type B, a type that is completely missing from Table \(\PageIndex{1}\).
The most luminous of the bright stars listed in Appendix J emit more than 50,000 times more energy than does the Sun. These highly luminous stars are missing from the solar neighborhood because they are very rare. None of them happens to be in the tiny volume of space immediately surrounding the Sun, and only this small volume was surveyed to get the data shown in Table \(\PageIndex{1}\).
For example, let’s consider the most luminous stars—those 100 or more times as luminous as the Sun. Although such stars are rare, they are visible to the unaided eye, even when hundreds to thousands of light-years away. A star with a luminosity 10,000 times greater than that of the Sun can be seen without a telescope out to a distance of 5000 light-years. The volume of space included within a distance of 5000 light-years, however, is enormous; so even though highly luminous stars are intrinsically rare, many of them are readily visible to our unaided eye.
The contrast between these two samples of stars, those that are close to us and those that can be seen with the unaided eye, is an example of a selection effect . When a population of objects (stars in this example) includes a great variety of different types, we must be careful what conclusions we draw from an examination of any particular subgroup. Certainly we would be fooling ourselves if we assumed that the stars visible to the unaided eye are characteristic of the general stellar population; this subgroup is heavily weighted to the most luminous stars. It requires much more effort to assemble a complete data set for the nearest stars, since most are so faint that they can be observed only with a telescope. However, it is only by doing so that astronomers are able to know about the properties of the vast majority of the stars, which are actually much smaller and fainter than our own Sun. In the next section, we will look at how we measure some of these properties.
Key Concepts and Summary
To understand the properties of stars, we must make wide-ranging surveys. We find the stars that appear brightest to our eyes are bright primarily because they are intrinsically very luminous, not because they are the closest to us. Most of the nearest stars are intrinsically so faint that they can be seen only with the aid of a telescope. Stars with low mass and low luminosity are much more common than stars with high mass and high luminosity. Most of the brown dwarfs in the local neighborhood have not yet been discovered.
Footnotes
1 The spectral types of stars were defined and discussed in Analyzing Starlight.
Glossary
- selection effect
- the selection of sample data in a nonrandom way, causing the sample data to be unrepresentative of the entire data set
Contributors and Attributions
-
Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


14.8: W. A. Mozart- His Works
Works, Musical Style, and Innovations
Style
Mozart’s music, like Haydn’s, stands as an archetype of the classical style. At the time he began composing, European music was dominated by the style galant , a reaction against the highly evolved intricacy of the baroque. Progressively, and in large part at the hands of Mozart himself, the contrapuntal complexities of the late baroque emerged once more, moderated and disciplined by new forms, and adapted to a new aesthetic and social milieu. Mozart was a versatile composer, and wrote in every major genre, including symphony, opera, the solo concerto, chamber music including string quartet and string quintet, and the piano sonata. These forms were not new, but Mozart advanced their technical sophistication and emotional reach. He almost single-handedly developed and popularized the classical piano concerto. He wrote a great deal of religious music, including large-scale masses, as well as dances, divertimenti, serenades, and other forms of light entertainment.
The central traits of the classical style are all present in Mozart’s music. Clarity, balance, and transparency are the hallmarks of his work, but simplistic notions of its delicacy mask the exceptional power of his finest masterpieces, such as the Piano Concerto No. 24 in C minor, K. 491; the Symphony No. 40 in G minor, K. 550; and the opera Don Giovanni . Charles Rosen makes the point forcefully:
It is only through recognizing the violence and sensuality at the center of Mozart’s work that we can make a start towards a comprehension of his structures and an insight into his magnificence. In a paradoxical way, Schumann’s superficial characterization of the G minor Symphony can help us to see Mozart’s daemon more steadily. In all of Mozart’s supreme expressions of suffering and terror, there is something shockingly voluptuous.
Especially during his last decade, Mozart exploited chromatic harmony to a degree rare at the time, with remarkable assurance and to great artistic effect.
Mozart always had a gift for absorbing and adapting valuable features of others’ music. His travels helped in the forging of a unique compositional language. In London as a child, he met J. C. Bach and heard his music. In Paris, Mannheim, and Vienna he met with other compositional influences, as well as the avant-garde capabilities of the Mannheim orchestra. In Italy he encountered the Italian overture and opera buffa, both of which deeply affected the evolution of his own practice. In London and Italy, the gallant style was in the ascendent: simple, light music with a mania for cadencing; an emphasis on tonic, dominant, and subdominant to the exclusion of other harmonies; symmetrical phrases; and clearly articulated partitions in the overall form of movements. Some of Mozart’s early symphonies are Italian overtures, with three movements running into each other; many are homotonal (all three movements having the same key signature, with the slow middle movement being in the relative minor). Others mimic the works of J. C. Bach, and others show the simple rounded binary forms turned out by Viennese composers.
Mozart would sometimes switch his focus between operas and instrumental music. He produced operas in each of the prevailing styles: opera buffa, such as The Marriage of Figaro , Don Giovanni , and Così fan tutte ; opera seria, such as Idomeneo ; and Singspiel, of which Die Zauberflöte is the most famous example by any composer. In his later operas he employed subtle changes in instrumentation, orchestral texture, and tone color, for emotional depth and to mark dramatic shifts. Here his advances in opera and instrumental composing interacted: his increasingly sophisticated use of the orchestra in the symphonies and concertos influenced his operatic orchestration, and his developing subtlety in using the orchestra to psychological effect in his operas was in turn reflected in his later non-operatic compositions.
Influence
Mozart’s most famous pupil, whom the Mozarts took into their Vienna home for two years as a child, was probably Johann Nepomuk Hummel, a transitional figure between classical and romantic eras. More important is the influence Mozart had on composers of later generations. Ever since the surge in his reputation after his death, studying his scores has been a standard part of the training of classical musicians.
Ludwig van Beethoven, Mozart’s junior by fifteen years, was deeply influenced by his work, with which he was acquainted as a teenager. He is thought to have performed Mozart’s operas while playing in the court orchestra at Bonn, and he traveled to Vienna in 1787 hoping to study with the older composer. Some of Beethoven’s works have direct models in comparable works by Mozart, and he wrote cadenzas (WoO 58) to Mozart’s D minor piano concerto K. 466. For further details see Mozart and Beethoven.
A number of composers have paid homage to Mozart by writing sets of variations on his themes. Beethoven wrote four such sets (Op. 66, WoO 28, WoO 40, WoO 46). Others include Fernando Sor’s Introduction and Variations on a Theme by Mozart (1821), Mikhail Glinka’s Variations on a Theme from Mozart’s Opera Die Zauberflöte (1822), Frédéric Chopin’s Variations on “Là ci darem la mano” from Don Giovanni (1827), and Max Reger’s Variations and Fugue on a Theme by Mozart (1914), based on the variation theme in the piano sonata K. 331.
Pyotr Ilyich Tchaikovsky wrote his Orchestral Suite No. 4 in G, “Mozartiana” (1887), as a tribute to Mozart.
Köchel Catalogue
For unambiguous identification of works by Mozart, a Köchel catalogue number is used. This is a unique number assigned, in regular chronological order, to every one of his known works. A work is referenced by the abbreviation “K.” or “KV” followed by this number. The first edition of the catalogue was completed in 1862 by Ludwig von Köchel. It has since been repeatedly updated, as scholarly research improves knowledge of the dates and authenticity of individual works.
Major Compositions
-
Operas:
- Lucio Silla, The Marriage of Figaro, Don Giovanni, The Magic Flute, Così fan tutte, and others
-
Vocal:
- Requiem, oratories, and cantatas
-
For orchestra:
- Symphonies, serenades, and divertimentos
- Concertos for different instruments and orchestra:
- More than thirty for piano and orchestra
- Five for violin and orchestra
- Four for horn and orchestra
-
Chamber Music:
- Six string quintets, more than twenty string quartets, more than thirty sonatas for violin and piano, trios, minuets, rondos, fantasies, and many other works
Contributors and Attributions
- Wolfgang Amadeus Mozart. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Wolfgang_Amadeus_Mozart . License : CC BY-SA: Attribution-ShareAlike
- Mozart. Authored by : Philip Edmondson. Located at : https://www.flickr.com/photos/philipedmondson/454040465/ . License : CC BY-NC: Attribution-NonCommercial
- Mozart Requiemu2014Karajan. Authored by : cagin. Located at : https://www.youtube.com/watch?v=ia8ceqIDSJw . License : All Rights Reserved . License Terms : Standard YouTube license
- Mozart Sheet Music. Provided by : Wikimedia. Located at : https://en.Wikipedia.org/wiki/Wolfgang_Amadeus_Mozart#/media/File:Mozart_Sheet_Music.jpg . License : Public Domain: No Known Copyright


11.5: Water Transport
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Water potential, transpiration, and stomatal regulation influence how water and nutrients are transported in plants. Water potential refers to the potential energy in water, and water moves towards the areas with the lowest water potential. Water is ultimately pulled to the top of the plant ( cohesion-tension theory ), and lost through transpiration through stomata. Complex mechanisms control stomatal opening and closure. Both adaptations that increase absorption of water through the roots (Figure \(\PageIndex{1}\)) and those that limit transpiration ensure that plants collect and retain enough water.
Curated and authored by Melissa Ha using 30.5 Transport of Water and Solutes in Plants from Biology 2e by OpenStax (licensed CC-BY ). Access for free at openstax.org .


4.3: The Mobius Function and the Mobius Inversion Formula
We start by defining the Mobius function which investigates integers in terms of their prime decomposition. We then determine the Mobius inversion formula which determines the values of the a function \(f\) at a given integer in terms of its summatory function.
\(\mu(n)=\left\{\begin{array}{lcr} \ 1 \ \ \mbox{if}\ \ n=1;\\ \ (-1)^t \ \ \mbox{if}\ \ n=p_1p_2...p_t \ \ \mbox{where the}\ \ p_i \ \ \mbox{are distinct primes};\\ \ 0 \ \ \mbox{ otherwise}.\\ \end{array}\right .\\)
Note that if \(n\) is divisible by a power of a prime higher than one then \(\mu(n)=0\).
In connection with the above definition, we have the following
An integer \(n\) is said to be square-free , if no square divides it, i.e. if there does not exist an integer \(k\) such that \(k^2\mid n\).
It is immediate (prove as exercise) that the prime-number factorization of a square-free integer contains only distinct primes.
Notice that \(\mu(1)=1\), \(\mu(2)=-1\), \(\mu(3)=-1\) and \(\mu(4)=0\).
We now prove that \(\mu(n)\) is a multiplicative function.
The Mobius function \(\mu(n)\) is multiplicative.
Let \(m\) and \(n\) be two relatively prime integers. We have to prove that \[\mu(mn)=\mu(m)\mu(n).\] If \(m=n=1\), then the equality holds. Also, without loss of generality, if \(m=1\), then the equality is also obvious. Now suppose that \(m\) or \(n\) is divisible by a power of prime higher than 1, then \[\mu(mn)=0=\mu(m)\mu(n).\] What remains to prove that if \(m\) and \(n\) are square-free integers say \(m=p_1p_2...p_s\) where \(p_1,p_2,...,p_s\) are distinct primes and \(n=q_1q_2...q_t\) where \(q_1,q_2,...,q_t\). Since \((m,n)=1\), then there are no common primes in the prime decomposition between \(m\) and \(n\). Thus \[\mu(m)=(-1)^s, \mu(n)=(-1)^t \mbox{and} \ \ \mu(mn)=(-1)^{s+t}.\]
In the following theorem, we prove that the summatory function of the Mobius function takes only the values \(0\) or \(1\).

For \(n=1\), we have \(F(1)=\mu(1)=1\). Let us now find \(\mu(p^k)\) for any integer \(k>0\). Notice that \[F(p^k)=\mu(1)+\mu(p)+...+\mu(p^k)=1+(-1)+0+...+0=0\] Thus by Theorem 36, for any integer \(n=p_1^{a_1}p_2^{a_2}...p_t^{a_t}>1\) we have, \[F(n)=F(p_1^{a_1})F(p_2^{a_2})...F(p_t^{a_t})=0\]
We now define the Mobius inversion formula. The Mobius inversion formula expresses the values of \(f\) in terms of its summatory function of \(f\).
Suppose that \(f\) is an arithmetic function and suppose that \(F\) is its summatory function, then for all positive integers \(n\) we have \[f(n)=\sum_{d\mid n}\mu(d)F(n/d).\]
We have \[\begin{aligned} \sum_{d\mid n}\mu(d)F(n/d)&=&\sum_{d\mid n}\mu(d)\sum_{e\mid (n/d)}f(e)\\ &=& \sum_{d\mid n}\sum_{e\mid (n/d)}\mu(d)f(e)\\&=&\sum_{e\mid n}\sum_{d\mid (n/e)}\mu(d)f(e)\\&=&\sum_{e\mid n}f(e)\sum_{d\mid (n/d)}\mu(d)\\\end{aligned}\] Notice that \(\sum_{d\mid (n/e)}\mu(d)=0\) unless \(n/e=1\) and thus \(e=n\). Consequently we get \[\sum_{e\mid n}f(e)\sum_{d\mid (n/d)}\mu(d)=f(n).1=f(n).\]
A good example of a Mobius inversion formula would be the inversion of \(\sigma(n)\) and \(\tau(n)\). These two functions are the summatory functions of \(f(n)=n\) and \(f(n)=1\) respectively. Thus we get \[n=\sum_{d\mid n}\mu(n/d)\sigma(d)\] and \[1=\sum_{d\mid n}\mu(n/d)\tau(d).\]
Exercises
- Find \(\mu(12)\) , \(\mu(10!)\) and \(\mu(105)\).
- Find the value of \(\mu(n)\) for each integer \(n\) with \(100\leq n\leq 110\).
- Use the Mobius inversion formula and the identity \(n=\sum_{d\mid n}\phi(n/d)\) to show that \(\phi(p^t)=p^t-p^{t-1}\) where \(p\) is a prime and \(t\) is a positive integer.
Contributors and Attributions
-
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


27.2A: Animal Characterization Based on Body Symmetry
- Differentiate among the ways in which animals can be characterized by body symmetry
Animal Characterization Based on Body Symmetry
At a very basic level of classification, true animals can be largely divided into three groups based on the type of symmetry of their body plan: radially symmetrical, bilaterally symmetrical, and asymmetrical. Only a few animal groups display radial symmetry, while asymmetry is a unique feature of phyla Porifera (sponges). All types of symmetry are well suited to meet the unique demands of a particular animal’s lifestyle.
Radial Symmetry
Radial symmetry is the arrangement of body parts around a central axis, like rays on a sun or pieces in a pie. Radially symmetrical animals have top and bottom surfaces, but no left and right sides, or front and back. The two halves of a radially symmetrical animal may be described as the side with a mouth (“oral side”) and the side without a mouth (“aboral side”). This form of symmetry marks the body plans of animals in the phyla Ctenophora (comb jellies) and Cnidaria (corals, sea anemones, and other jellies). Radial symmetry enables these sea creatures, which may be sedentary or only capable of slow movement or floating, to experience the environment equally from all directions.
Bilateral Symmetry
Bilateral symmetry involves the division of the animal through a sagittal plane, resulting in two mirror-image, right and left halves, such as those of a butterfly, crab, or human body. Animals with bilateral symmetry have a “head” and “tail” (anterior vs. posterior), front and back (dorsal vs. ventral), and right and left sides. All true animals, except those with radial symmetry, are bilaterally symmetrical. The evolution of bilateral symmetry and, therefore, the formation of anterior and posterior (head and tail) ends promoted a phenomenon called cephalization, which refers to the collection of an organized nervous system at the animal’s anterior end. In contrast to radial symmetry, which is best suited for stationary or limited-motion lifestyles, bilateral symmetry allows for streamlined and directional motion. In evolutionary terms, this simple form of symmetry promoted active mobility and increased sophistication of resource-seeking and predator-prey relationships.
Animals in the phylum Echinodermata (such as sea stars, sand dollars, and sea urchins) display radial symmetry as adults, but their larval stages exhibit bilateral symmetry. This is termed secondary radial symmetry. They are believed to have evolved from bilaterally symmetrical animals; thus, they are classified as bilaterally symmetrical.
Asymmetry
Only members of the phylum Porifera (sponges) have no body plan symmetry. There are some fish species, such as flounder, that lack symmetry as adults. However, the larval fish are bilaterally symmetrical.
Key Points
- Animals with radial symmetry have no right or left sides, only a top or bottom; these species are usually marine organisms like jellyfish and corals.
- Most animals are bilaterally symmetrical with a line of symmetry dividing their body into left and right sides along with a “head” and “tail” in addition to a top and bottom.
- Only sponges (phylum Porifera) have asymmetrical body plans.
- Some animals start life with one type of body symmetry, but develop a different type as adults; for example, sea stars are classified as bilaterally symmetrical even though their adult forms are radially symmetrical.
Key Terms
- sagittal plane : divides the body into right and left halves
- radial symmetry : a form of symmetry wherein identical parts are arranged in a circular fashion around a central axis
- bilateral symmetry : having equal arrangement of parts (symmetry) about a vertical plane running from head to tail


10.14: The Volatility of the Time Value of Money
Discrepancies in TVM factors will widen as time increases, as one observes the relative factors between interest rate columns.
For example, a five-year IOU with a future value of $1,000, using the tables, would have a present value of $1,000 (0.7835) = $783.50 – at a discount rate of 5%. The IOU could be purchased or sold for that amount, or price. Think of present value as an item’s dollar price. If the discount rate instead were 10%, the present value would be only: $1,000 (0.6209) = $620.90. In percentage terms, the present value of $1,000 to be received five years from now, discounted at a rate of 5% is greater than at 10% by a difference of (783.5 ÷ 620.9) – 1 = 26.2%.
If however the IOU had a 30-year term, the difference in present value would itself compound. At 5%, the present value would equal $1,000 (0.2314) = $231.40. At 10%, the PV would equal $1,000 (0.0573) = $57.30. In percentage terms, the present value of $1,000 to be received thirty years from now, discounted at a rate of 5% is greater than at 10% by a difference of (231.4 ÷ 57.30) – 1 = 303.8% .
This demonstrates the volatility and geometry of TVM! By geometry here we refer to its non-linear and exponential nature. Whenever there is an exponent in a formula, we get some kind of curve . As time increases, differences in present- and future-values for a given number of years themselves increase non-linearly.
If you had purchased a thirty-year IOU as an investment, any changes in interest rates (i.e., due to market conditions) would have a far greater impact on the value of your IOU investment than if you had, instead, purchased a five-year obligation. For a given change in discount rates of interest, the impact on the multipliers is greater the greater the time is. T he impact on price, which is present value, is greater, the greater the time – period . “Price volatility,” so to speak, increases as the future payment grows more distant.
Again, this is because, the time value of money is non-linear; it is exponential. We are dealing, quite literally, with compound interest, i.e., interest on the interest. Holders of long-term fixed obligations, such as bonds, may experience greater price, or market value fluctuations, when discount rates for their bonds suddenly change.
Bonus Question : In the example above, we examined the increase in the Present Value Factor when interest rates drop from 10% to 5%. What would be the percentage change in the Factors if rates increased from 5% to 10%? Would it be same percentage change?
The virtually instantaneous changes in present values , when going from a discount rate of 10% to 5% , increases (“first derivative”) at a decreasing rate (“second derivative”). The table displays the extent to which 5% discounted present values exceed 10% discounted values.
When going from 10% to 5%, a five-year payment will increase in value about 26%, while a thirty-year payment by over 300%! Imagine if you could buy an IOU at 10% and immediately (“instantaneously”) turn around and sell it at 5% ! Your profit would be much greater had you invested in the thirty-year obligation. While this case is exaggerated, the bond market works in similar fashion. Bond pr i ces can, at times, be quite volatile due to changes in market rates although large changes do not occur instantaneously) except in the case of a disaster). Remember : prices are the present values of a bond’s future payments !
These relationships can also be illustrated using Differential Calculus, which would give you a more “continuous,” rather than a “discrete,” view of the progress of the numbers.
The Take-away : If interest rates change, (bond) prices could change dramatically!!


4.10: Add and Subtract Mixed Numbers (Part 1)
- Model addition of mixed numbers with a common denominator
- Add mixed numbers with a common denominator
- Model subtraction of mixed numbers
- Subtract mixed numbers with a common denominator
- Add and subtract mixed numbers with different denominators
Before you get started, take this readiness quiz.
- Draw figure to model \(\dfrac{7}{3}\). If you missed this problem, review Example 4.1.6 .
- Change \(\dfrac{11}{4}\) to a mixed number. If you missed this problem, review Example 4.1.9 .
- Change \(3 \dfrac{1}{2}\) to an improper fraction. If you missed this problem, review Example 4.1.11 .
Model Addition of Mixed Numbers with a Common Denominator
So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money.
If Ron has \(1\) dollar and \(1\) quarter, he has \(1 \dfrac{1}{4}\) dollars. If Don has \(2\) dollars and \(1\) quarter, he has \(2 \dfrac{1}{4}\) dollars. What if Ron and Don put their money together? They would have \(3\) dollars and \(2\) quarters. They add the dollars and add the quarters. This makes \(3 \dfrac{2}{4}\) dollars. Because two quarters is half a dollar, they would have \(3\) and a half dollars, or \(3 \dfrac{1}{2}\) dollars.
\[\begin{split} & 1 \dfrac{1}{4} \\ + & 2 \dfrac{1}{4} \\ \hline \\ & 3 \dfrac{2}{4} = 3 \dfrac{1}{2} \end{split} \nonumber \]
When you added the dollars and then added the quarters, you were adding the whole numbers and then adding the fractions.
\[1 \dfrac{1}{4} + 2 \dfrac{1}{4} \nonumber \]
We can use fraction circles to model this same example:
| Start with \(1 \dfrac{1}{4}\). | one whole and one \(\dfrac{1}{4}\) pieces | \(1 \dfrac{1}{4}\) | |
| Add \(2 \dfrac{1}{4}\) more. | two wholes and one \(\dfrac{1}{4}\) pieces | \(\begin{split}+ & 2 \dfrac{1}{4} \\ & \hline \end{split}\) | |
| The sum is: | three wholes and two \(\dfrac{1}{4}\)'s | \(3 \dfrac{2}{4} = 3 \dfrac{1}{2}\) |
Model \(2 \dfrac{1}{3} + 1 \dfrac{2}{3}\) and give the sum.
Solution
We will use fraction circles, whole circles for the whole numbers and \(\dfrac{1}{3}\) pieces for the fractions.
| two wholes and one \(\dfrac{1}{3}\) | \(2 \dfrac{1}{3}\) | |
| plus one whole and two \(\dfrac{1}{3}\)s | \(\begin{split}+ & 1 \dfrac{2}{3} \\ & \hline \end{split}\) | |
| sum is three wholes and three \(\dfrac{1}{3}\)s | \(3 \dfrac{3}{3} = 4\) |
This is the same as \(4\) wholes. So, \(2 \dfrac{1}{3} + 1 \dfrac{2}{3} = 4\).
Use a model to add the following. Draw a picture to illustrate your model. \(1 \dfrac{2}{5} + 3 \dfrac{3}{5}\)
- Answer
-
\(5\)
Use a model to add the following. Draw a picture to illustrate your model. \(2 \dfrac{1}{6} + 2 \dfrac{5}{6}\)
- Answer
-
\(5\)
Model \(1 \dfrac{3}{5} + 2 \dfrac{3}{5}\) and give the sum as a mixed number.
Solution
We will use fraction circles, whole circles for the whole numbers and \(\dfrac{1}{5}\) pieces for the fractions.
| one whole and three \(\dfrac{1}{5}\)s | \(1 \dfrac{3}{5}\) | |
| plus two wholes and three \(\dfrac{1}{5}\)s | \(\begin{split}+ & 2 \dfrac{3}{5} \\ & \hline \end{split}\) | |
| sum is three wholes and six \(\dfrac{1}{5}\)s | \(3 \dfrac{6}{5} = 4 \dfrac{1}{5}\) |
Adding the whole circles and fifth pieces, we got a sum of \(3 \dfrac{6}{5}\). We can see that \(\dfrac{6}{5}\) is equivalent to \(1 \dfrac{1}{5}\), so we add that to the \(3\) to get \(4 \dfrac{1}{5}\).
Model, and give the sum as a mixed number. Draw a picture to illustrate your model. \(2 \dfrac{5}{6} + 1 \dfrac{5}{6}\)
- Answer
-
\(4\dfrac{2}{3}\)
Model, and give the sum as a mixed number. Draw a picture to illustrate your model. \(1 \dfrac{5}{8} + 1 \dfrac{7}{8}\)
- Answer
-
\(3\dfrac{1}{2}\)
Add Mixed Numbers
Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.
Step 1. Add the whole numbers.
Step 2. Add the fractions.
Step 3. Simplify, if possible.
Add: \(3 \dfrac{4}{9} + 2 \dfrac{2}{9}\).
Solution
| Add the whole numbers. | \(\begin{split} & \textcolor{red}{3} \dfrac{4}{9} \\ + & \textcolor{red}{2} \dfrac{2}{9} \\ \hline \\ & \textcolor{red}{5} \end{split}\) |
| Add the fractions. | \(\begin{split} & 3 \textcolor{red}{\dfrac{4}{9}} \\ + & 2 \textcolor{red}{\dfrac{2}{9}} \\ \hline \\ & 5 \textcolor{red}{\dfrac{6}{9}} \end{split}\) |
| Simplify the fraction. | \(\begin{split} & 3 \dfrac{4}{9} \\ + & 2 \dfrac{2}{9} \\ \hline \\ & \textcolor{red}{5 \dfrac{6}{9}} = 5 \dfrac{2}{3} \end{split}\) |
Find the sum: \(4 \dfrac{4}{7} + 1 \dfrac{2}{7}\).
- Answer
-
\(5\dfrac{6}{7}\)
Find the sum: \(2 \dfrac{3}{11} + 5 \dfrac{6}{11}\).
- Answer
-
\(7\dfrac{9}{11}\)
In Example \(\PageIndex{3}\), the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.
Find the sum: \(9 \dfrac{5}{9} + 5 \dfrac{7}{9}\).
Solution
| Add the whole numbers and then add the fractions. | \(\begin{split} & 9 \dfrac{5}{9} \\ + & 5 \dfrac{7}{9} \\ \hline \\ & 14 \dfrac{12}{9} \end{split}\) |
| Rewrite \(\dfrac{12}{9}\) as an improper fraction. | \(14 + 1 \dfrac{3}{9}\) |
| Add. | \(15 \dfrac{3}{9}\) |
| Simplify. | \(15 \dfrac{1}{3}\) |
Find the sum: \(8 \dfrac{7}{8} + 7 \dfrac{5}{8}\).
- Answer
-
\(16\dfrac{1}{2}\)
Find the sum: \(6 \dfrac{7}{9} + 8 \dfrac{5}{9}\).
- Answer
-
\(15\dfrac{1}{3}\)
An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.
Add by converting the mixed numbers to improper fractions: \(3 \dfrac{7}{8} + 4 \dfrac{3}{8}\).
Solution
| Convert to improper fractions. | \(\dfrac{31}{8} + \dfrac{35}{8}\) |
| Add the fractions. | \(\dfrac{31 + 35}{8}\) |
| Simplify the numerator. | \(\dfrac{66}{8}\) |
| Rewrite as a mixed number. | \(8 \dfrac{2}{8}\) |
| Simplify the fraction. | \(8 \dfrac{1}{4}\) |
Since the problem was given in mixed number form, we will write the sum as a mixed number.
Find the sum by converting the mixed numbers to improper fractions: \(5 \dfrac{5}{9} + 3 \dfrac{7}{9}\)
- Answer
-
\(9\dfrac{1}{3}\)
Find the sum by converting the mixed numbers to improper fractions: \(3 \dfrac{7}{10} + 2 \dfrac{9}{10}\)
- Answer
-
\(6\dfrac{3}{5}\)
Table \(\PageIndex{1}\) compares the two methods of addition, using the expression \(3 \dfrac{2}{5} + 6 \dfrac{4}{5}\) as an example. Which way do you prefer?
| Mixed Numbers | Improper Fractions |
|---|---|
| \(\begin{split} & 3 \dfrac{2}{5} \\ + & 6 \dfrac{4}{5} \\ \hline \\ & 9 \dfrac{6}{5} \end{split}\) | \(3 \dfrac{2}{5} + 6 \dfrac{4}{5}\) |
| \(9 + \dfrac{6}{5}\) | \(\dfrac{17}{5} + \dfrac{34}{5}\) |
| \(9 + 1 \dfrac{1}{5}\) | \(\dfrac{51}{5}\) |
| \(10 \dfrac{1}{5}\) | \(10 \dfrac{1}{5}\) |
Model Subtraction of Mixed Numbers
Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half.
We will use fraction circles (pizzas!) to help us visualize the process. Start with one whole.
Figure \(\PageIndex{1}\)
Algebraically, you would write:
Use a model to subtract: \(1 − \dfrac{1}{3}\).
Solution
Use a model to subtract: \(1 − \dfrac{1}{4}\).
- Answer
-
\(\dfrac{3}{4}\)
Use a model to subtract: \(1 − \dfrac{1}{5}\).
- Answer
-
\(\dfrac{4}{5}\)
What if we start with more than one whole? Let’s find out.
Use a model to subtract: \(2 − \dfrac{3}{4}\).
Solution
Use a model to subtract: \(2 − \dfrac{1}{5}\).
- Answer
-
\(\dfrac{9}{5}\)
Use a model to subtract: \(2 − \dfrac{1}{3}\).
- Answer
-
\(\dfrac{5}{3}\)
In the next example, we’ll subtract more than one whole.
Use a model to subtract: \(2 − 1 \dfrac{2}{5}\).
Solution
Use a model to subtract: \(2 − 1 \dfrac{1}{3}\).
- Answer
-
\(\dfrac{2}{3}\)
Use a model to subtract: \(2 − 1 \dfrac{1}{4}\).
- Answer
-
\(\dfrac{3}{4}\)
What if you start with a mixed number and need to subtract a fraction? Think about this situation: You need to put three quarters in a parking meter, but you have only a \(\$1\) bill and one quarter. What could you do? You could change the dollar bill into \(4\) quarters. The value of \(4\) quarters is the same as one dollar bill, but the \(4\) quarters are more useful for the parking meter. Now, instead of having a \(\$1\) bill and one quarter, you have \(5\) quarters and can put \(3\) quarters in the meter.
This models what happens when we subtract a fraction from a mixed number. We subtracted three quarters from one dollar and one quarter.
We can also model this using fraction circles, much like we did for addition of mixed numbers.
Use a model to subtract: \(1 \dfrac{1}{4} − \dfrac{3}{4}\)
Solution
| Rewrite vertically. Start with one whole and one fourth. | \(\begin{split} & \textcolor{red}{1 \dfrac{1}{4}} \\ - & \dfrac{3}{4} \\ \hline \end{split}\) | |
| Since the fractions have denominator 4, cut the whole into 4 pieces. You now have \(\dfrac{4}{4}\) and \(\dfrac{1}{4}\) which is \(\dfrac{5}{4}\). | \(\begin{split} & \textcolor{red}{\dfrac{5}{4}} \\ - & \dfrac{3}{4} \\ \hline \end{split}\) | |
| Take away \(\dfrac{3}{4}\). There is \(\dfrac{1}{2}\) left. | \(\begin{split} & \dfrac{5}{4}\\\textcolor{red}{- & \dfrac}3}{4}}\\ \hline \\ & \dfrac{2}{4} = \dfrac{1}{2} \end{split}\) |
Use a model to subtract. Draw a picture to illustrate your model. \(1 \dfrac{1}{3} − \dfrac{2}{3}\)
- Answer
-
Use a model to subtract. Draw a picture to illustrate your model. \(1 \dfrac{1}{5} − \dfrac{4}{5}\)
- Answer
-
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


3.3: The Extended Plane
Consider again inversion about the circle \(C\) given by \(|z - z_0| = r\text{,}\) and observe that points close to \(z_0\) get mapped to points in the plane far away from \(z_0\text{.}\) In fact, a sequence of points in \(\mathbb{C}\) whose limit is \(z_0\) will be inverted to a sequence of points whose magnitudes go to \(\infty\text{.}\) Conversely, any sequence of points in \(\mathbb{C}\) having magnitudes marching off to \(\infty\) will be inverted to a sequence of points whose limit is \(z_0\text{.}\)
With this in mind, we define a new point called the point at infinity , denoted \(\infty.\) Adjoin this new point to the plane to get the extended plane , denoted as \({\mathbb{C}}^+\text{.}\) Then, one may extend inversion in the circle \(C\) to include the points \(z_0\) and \(\infty\text{.}\) In particular, inversion of \(\mathbb{C}^+\) in the circle \(C\) centered at \(z_0\) with radius \(r\text{,}\) \(i_C: \mathbb{C}^+ \to \mathbb{C}^+\text{,}\) is given by
\[ i_C(z) = \begin{cases}\dfrac{r^2}{(\overline{z-z_0})} + z_0 & \text{ if \(z \neq z_0,\infty\); } \\ \infty & \text{ if \(z=z_0\); } \\ z_0 & \text{ if \(z = \infty\) } \end{cases}\text{.} \]
Viewing inversion as a transformation of the extended plane, we define \(z_0\) and \(\infty\) to be symmetric points with respect to the circle of inversion.
The space \({\mathbb{C}}^+\) will be the canvas on which we do all of our geometry, and it is important to begin to think of \(\infty\) as “one of the gang,” just another point to consider. All of our translations, dilations, and rotations can be redefined to include the point \(\infty\text{.}\)
So where is \(\infty\) in \({\mathbb{C}}^+\text{?}\) You approach \(\infty\) as you proceed in either direction along any line in the complex plane. More generally, if \(\{z_n\}\) is a sequence of complex numbers such that \(|z_n| \to \infty\) as \(n \to \infty\text{,}\) then we say \(\displaystyle\lim_{n\to\infty} z_n = \infty\text{.}\) By convention, we assume \(\infty\) is on every line in the extended plane, and reflection across any line fixes \(\infty\text{.}\)
Any general linear transformation extended to the domain \({\mathbb{C}}^+\) fixes \(\infty\text{.}\)
- Proof
-
If \(T(z) = a z + b\) where \(a\) and \(b\) are complex constants with \(a \neq 0\text{,}\) then by limit methods from calculus, as \(|z_n| \to \infty\text{,}\) \(|a z_n + b| \to \infty\) as well. Thus, \(T(\infty) = \infty\text{.}\)
So, with new domain \({\mathbb{C}}^+\text{,}\) we modify our fixed point count for the basic transformations:
- The translation \(T_b\) of \({\mathbb{C}}^+\) fixes one point (\(\infty\)).
- The rotation about the origin \(R_\theta\) of \({\mathbb{C}}^+\) fixes \(2\) points (\(0\) and \(\infty\)).
- The dilation \(T(z) = kz\) of \({\mathbb{C}}^+\) fixes \(2\) points, (\(0\) and \(\infty\)).
- The reflection \(r_L(z)\) of \({\mathbb{C}}^+\) about line \(L\) fixes all points on \(L\) (which now includes \(\infty\)).
The following function is a transformation of \({\mathbb{C}}^+\)
\[ T(z) = \dfrac{i+1}{z+2i}\text{,} \]
a fact we prove in the next section. For now, we ask where \(T\) sends \(\infty\text{,}\) and which point gets sent to \(\infty\text{.}\)
We tackle the second question first. The input that gets sent to \(\infty\) is the complex number that makes the denominator 0. Thus, \(T(-2i) = \infty.\)
To answer the first question, take your favorite sequence that marches off to \(\infty\text{,}\) for example, \(1, 2, 3,\ldots\text{.}\) The image of this sequence, \(T(1), T(2),\) \(T(3),\ldots\) consists of complex fractions in which the numerator is constant, but the denominator grows unbounded in magnitude along the horizontal line \(\text{Im}(z) = 2\text{.}\) Thus, the quotient tends to \(0\), and \(T(\infty) = 0\text{.}\)
As a second example, you can check that if
\[ T(z) = \dfrac{iz+(3i+1)}{2iz+1}, \]
then \(T(\dfrac{i}{2}) = \infty\) and \(T(\infty) = \dfrac{1}{2}\text{.}\)
We emphasize that the following key results of the previous section extend to \(\mathbb{C}^+\) as well:
- There exists a unique cline through any three distinct points in \(\mathbb{C}^+\text{.}\) (If one of the given points in Theorem \(3.2.1\) is \(\infty\text{,}\) the unique cline is the line through the other two points.)
- Theorem \(3.2.3\) applies to all points \(z\) not on \(C\text{,}\) including \(z = z_0\) or \(\infty\text{.}\)
- Inversion about a cline preserves angle magnitudes at all points in \(\mathbb{C}^+\) (we discuss this below).
- Inversion preserves symmetry points for all points in \(\mathbb{C}^+\) (Theorem \(3.2.5\) holds if \(p\) or \(q\) is \(\infty\)).
- Theorem \(3.2.7\) now holds for all clines that do not intersect, including concentric circles. If the circles are concentric, the points symmetric to both of them are \(\infty\) and the common center.
We close this section with a look at stereographic projection. By identifying the extended plane with a sphere, this map offers a very useful way for us to think about the point \(\infty\text{.}\)
The unit \(2\)-sphere , denoted \(\mathbb{S}^2\text{,}\) consists of all the points in \(3\)-space that are one unit from the origin. That is,
\[ \mathbb{S}^2 = \{(a,b,c) \in \mathbb{R}^3 ~|~ a^2+b^2+c^2=1\}\text{.} \]
We will usually refer to the unit \(2\)-sphere as simply “the sphere.” Stereographic projection of the sphere onto the extended plane is defined as follows. Let \(N = (0,0,1)\) denote the north pole on the sphere. For any point \(P \neq N\) on the sphere, \(\phi(P)\) is the point on the ray \(\overrightarrow{NP}\) that lives in the \(xy\)-plane. See Figure \(3.3.1\) for the image of a typical point \(P\) of the sphere.
The stereographic projection map \(\phi\) can be described algebraically. The line through \(N = (0,0,1)\) and \(P = (a,b,c)\) has directional vector \(\overrightarrow{NP}=\langle a,b,c-1\rangle\text{,}\) so the line equation can be expressed as
\[ {\vec r}(t) = \langle 0,0,1\rangle + t\langle a,b,c-1\rangle\text{.} \]
This line intersects the \(xy\)-plane when its \(z\) coordinate is zero. This occurs when \(t = \dfrac{1}{1-c}\text{,}\) which corresponds to the point \((\dfrac{a}{1-c}, \dfrac{b}{1-c},0)\text{.}\)
Thus, for a point \((a,b,c)\) on the sphere with \(c \neq 1\text{,}\) stereographic projection \(\phi:\mathbb{S}^2 \to \mathbb{C}^+\) is given by
\[ \phi((a,b,c)) = \dfrac{a}{1-c}+\dfrac{b}{1-c}i\text{.} \]
Where does \(\phi\) send the north pole? To \(\infty\text{,}\) of course. A sequence of points on \(\mathbb{S}^2\) that approaches \(N\) will have image points in \(\mathbb{C}\) with magnitudes that approach \(\infty\text{.}\)
If we think of \(\infty\) as just another point in \(\mathbb{C}^+\text{,}\) it makes sense to ask about angles at this point. For instance, any two lines intersect at \(\infty\text{,}\) and it makes sense to ask about the angle of intersection at \(\infty\text{.}\) We can be guided in answering this question by stereographic projection, thanks to the following theorem.
Stereographic projection preserves angles. That is, if two curves on the surface of the sphere intersect at angle \(\theta\text{,}\) then their image curves in \(\mathbb{C}^+\) also intersect at angle \(\theta\text{.}\)
Thus, if two curves in \(\mathbb{C}^+\) intersect at \(\infty\) we may define the angle at which they intersect to equal the angle at which their pre-image curves under stereographic projection intersect. The angle at which two parallel lines intersect at \(\infty\) is 0. Furthermore, if two lines intersect at a finite point \(p\) as well as at \(\infty\text{,}\) the angle at which they intersect at \(\infty\) equals the negative of the angle at which they intersect at \(p\text{.}\) As a consequence, we may say that inversion about a circle preserves angle magnitudes at all points in \(\mathbb{C}^+\text{.}\)
Exercises
In each case find \(T(\infty)\) and the input \(z_0\) such that \(T(z_0) = \infty\text{.}\)
- \(T(z) = \dfrac{(3 - z)}{(2z + i)}\text{.}\)
- \(T(z) = \dfrac{(z + 1)}{e^{i\pi/4}}\text{.}\)
- \(T(z) = \dfrac{(az + b)}{(cz + d)}\text{.}\)
Suppose \(D\) is a circle of Apollonius of \(p\) and \(q\text{.}\) Prove that \(p\) and \(q\) are symmetric with respect to \(D\text{.}\)
- Hint
-
Recall the circle \(C\) in the proof of Theorem \(3.2.6\). Show that \(p\) and \(q\) get sent to points that are symmetric with respect to \(i_C(D)\text{.}\)
Determine the inverse stereographic projection function \(\phi^{-1}:\mathbb{C}^+ \to \mathbb{S}^2\text{.}\) In particular, show that for \(z = x + yi \neq \infty\text{,}\)
\[ \phi^{-1}(x,y) = \bigg(\dfrac{2x}{x^2+y^2+1},\dfrac{2y}{x^2+y^2+1},\dfrac{x^2+y^2-1}{x^2+y^2+1}\bigg). \]


13.9: Summary
Career Connection
Your Best Self
Returning to our definition of self-care – self-care is anything that makes you better – there is an addition that may help direct you in further developing your approach. Humanist Psychologist Carl Rogers described the healthy and vital individual as a fully functioning person. According to Rogers, this person strives to and may act in congruence with what they envisioned as their ideal, best self. This was the process that Rogers referred to as self-actualization, and as a precondition to authentically helping others. Based on this understanding, a revised definition of self-care could be: self-care is anything that makes you better and helps you live more like your best self .
Previously mentioned was the idea that self-care says, “you can’t pour from an empty cup.” If self-care is, in part, about realizing our best selves, it matters what you fill your cup with. There may be times when getting food-to-go and watching Netflix is the best thing for your mind, body, and soul. Other times, those same things could just be more junk that isn’t serving you or your development towards your best self. Therefore, when you think about your self-care needs, it may be crucial to ask:
- “What does my best self want?”
- “What does my best self need?”
- “What can I do now that my current self and future self will both thank me for?”
To help guide you in recognizing and acting in alignment with your ideal self, self-awareness and self-compassion may be essential. The next section covers these in more detail.
ACTIVITY : MY BEST SELF
My Best Self
Describe your best self. At your best, according to you:
- What is your best physical self? What can you do with your physical body? How do you feel?
- What is your best mental self? What do you think? What is your mindset? How do you care for a healthy mind?
- What is your best emotional self? What do you to support positive emotional states? How do care for yourself in negative emotional states? How do you develop greater emotional intelligence?
- What is your best social self? What are your social networks? Who are your most important relationships? How do you nurture them?
Find a time to give yourself 20 minutes or more of uninterrupted writing.
Key Takeaways
- Sexual health is an important dimension of wellness and something we should all think about to affirm our values and make responsible decisions.
- Your time in college and your overall health and well-being would be seriously impacted if you were to acquire a sexually transmitted infection or experience an unwanted pregnancy. You owe it to yourself—and anyone with whom you are in a relationship—to have the facts and know how to protect yourself.
- The huge number of sexual assaults that occur every year is one of our society’s “dirty little secrets.” This problem is as rampant on college campuses as in society in general. You need to know what’s involved—and what to do to protect yourself from the pain of becoming a victim.
- Good health helps you be more successful in college.
- For good nutrition, eat a varied diet with lots of vegetables, fruits, and whole grains and minimize fats, sugar, and salt.
- Regular exercise is not only important for good health but is a great way to reduce stress in your life.
- Sleep is one of the first areas where college students cut back when they find themselves too busy with classes, work, and other activities. Taking the time to get enough sleep, however, makes you so much more efficient when studying that it can actually save you time.
- Substance use and abuse not only takes its toll on the body but also contributes to problems in college, at work, and in the future. You may need to make a smart decision between short-term pleasures and long-term success.
- Since many stressors are unavoidable in life, we all need to find good ways to minimize their effects. The best stress-reducers over time become good habits that will increase our wellness and help us succeed in college and careers.
- If you are having an emotional or relationship problem that persists and affects your life, don’t hesitate to seek help. Most colleges have counselors and health professionals trained to help you get through any crisis.
- Sexual health is your own business—except that sexuality usually affects and is affected by others. Smart choices focus on protecting yourself from potential problems, regardless of your choices about sexual activity.


3.2: Sexuality and Intimacy
Understanding Healthy Relationships
Watch these two short videos:
Are you satisfied with the depths of the relationships in your life?
Sexuality and Intimacy
The idea that sex can be an addiction is new to many people.
The term “addiction” has become a popular metaphor to describe any form of self-destructive behavior that one is unable to stop despite known and predictable adverse consequences.  It involves frequent self-destructive or high risk activity that is not emotionally fulfilling, that one is ashamed of, and that one is unable to stop despite it causing repeated problems in the areas of marriage, social relationships, health, employment, finances, or the law.
Recognition that self-destructive sexual behavior can be an addiction has spawned the rapid growth of four nationwide self-help organizations for persons trying to recover from this problem. All are 12-step recovery programs patterned after Alcoholics Anonymous.
One might ask how sex can be an addiction when it is doing what comes naturally and does not involve abuse of a psychoactive substance like drugs or alcohol. The scientific argument for addiction is based, in part, on recent advances in neurochemistry that suggest we carry within us our own source of addictive chemicals.
When pleasure centers in the human brain are stimulated, chemicals called endorphins are released into the blood stream.  Any chemical that causes mood changes can be addictive, with repeated exposure altering brain chemistry to the point that more of the chemical is “required” in order to feel “normal.”
For example, experiments with hamsters have shown that the level of endorphins in their blood increases dramatically after several ejaculations. Experimental rats habituated to endorphins will go through much pain in order to obtain more. In rats, the addiction to endorphins is even stronger than the addiction to morphine or heroin.
The sex addict uses sex as a quick fix, or as a form of medication for anxiety, pain, loneliness, stress, or sleep. Sex addicts often refer to sex as their “pain reliever” or “tension reliever.” In a popular novel, the heroine describes sex as “the thinking women’s Valium.”

Compulsive or addictive sexual behavior may take various forms, including what many regard as “normal” heterosexual behavior. The type of sexual activity and even the frequency or number of partners are not of great significance in diagnosing this problem.  What is significant is a pattern of self-destructive or high risk sexual behavior that is unfulfilling and that a person is unable to stop.
The roots of out-of-control sexual behavior may be quite varied. It may be caused by an underlying personality disorder, an “addiction” to sex, or a physical disorder. The traditional disorders of exaggerated sexuality, nymphomania in the female and satyriasis in the male, are believed to be caused by a disorder of the pituitary gland or irritation of the brain cortex by a tumor, arteriosclerosis or epilepsy. These physical disorders are rare.
Compulsive or addictive sexual behavior is a concern because it may lead to poor judgment or lack of discretion, indicate a serious emotional or mental problem, open one to exploitation, manipulation, or extortion.
Sources
Sexuality and Intimacy: Sexual Addiction , USDA, http://www.dm.usda.gov/ocpm/Security%20Guide/Eap/Sex.htm


8.6: Interpreting the y-Intercept of a Line
Learning Outcomes
- Interpret the \(y\)-intercept of a line as the value of \(y\) when \(x\) equals 0.
- Determine whether the \(y\)-intercept is useful for interpreting the relationship between \(x\) and \(y\).
Just like the slope of a line, many algebra classes go over the \(y\)-intercept of a line without explaining how to use it in the real world. The \(y\)-intercept of a line is the value of \(y\) where the line crosses the \(y\)-axis. In other words, it is the value of \(y\) when the value of \(x\) is equal to 0. Sometimes this has true meaning for the model that the line provides, but other times it is meaningless. We will encounter examples of both types in this section.
Interpreting the \(y\)-Intercept of a Line
When the value for the \(x\)-variable is 0, the best prediction for the value of the \(y\)-variable is the value of the \(y\)-intercept.
Example \(\PageIndex{1}\)
A study was done to see the relationship between the ounces of meat, \(x\), that people eat each day on average and the hours per week, \(y\) they watch sports. The equation of the regression line was found to be:
\[y=1.3\:+0.4x\nonumber \]
Interpret the \(y\)-intercept of the regression line in the context of the study or explain why it has no practical meaning.
Solution
First, note that the \(y\)-intercept is the number that is not in front of the \(x\). Thus, the \(y\)-intercept is 1.3. Next, the \(y\)-intercept is the value of \(y\) when \(x\) equals zero. For this example, \(x\) represents the ounces of meat consumed each day.
When the consumption of meat is 0 ounces, the best prediction for the value of the hours of sports watched each week is 1.3.
If \(x\) is equal to 0, this means the person does not consume any meat. Since there are people, namely vegetarians or vegans, who consume no meat, it is meaningful to have an \(x\)-value of 0. The \(y\)-value of 1.3 represents the hours of sports the person watches per week. Putting this all together we can state:
A person who does not eat meat (vegetarian/vegan) is predicted to watch 1.3 hours of sports each week.
Example \(\PageIndex{2}\)
A neonatal nurse at Children's Hospital has collected data on the birth weight, \(x\), in pounds and the number of days, \(y\), that the newborns stay in the hospital. The equation of the regression line was found to be
\[y=45\:-3.9x\nonumber \]
Interpret the \(y\)-intercept of the regression line in the context of the study or explain why it has no practical meaning.
Solution
Again, we note that the \(y\)-intercept is the number that is not in front of the \(x\). Thus, the y-intercept is 45. Next, the \(y\)-intercept is the value of \(y\) when \(x\) equals zero.
When the birth weight in pounds is 0, the best prediction for the value of the number of days the newborn is predicted to stay in the hospital is 45 days.
For this example, \(x\) represents the new born baby's birth weight in pounds. If \(x\) is equal to 0, this means the baby was born with a weight of 0 pounds. Since it makes no sense for a baby to weigh 0 pounds, we can say that the \(y\) -intercept of this regression line has no practical meaning.
Example \(\PageIndex{3}\)
A researcher asked several people "How many cups of coffee did you drink last week?" and "How many times did you go to a shop or restaurant for a meal or a drink last week?" The scatterplot and the regression line from this study are shown below.
Interpret the \(y\)-intercept of the regression line in the context of the study or explain why it has no practical meaning.
Solution
The \(y\)-intercept of a line is where it crosses the \(y\)-axis. In this case, the line crosses at around \(y\) = -1. The value of \(x\), by definition is 0 and the \(x\)-axis represents the number of cups of coffee a person drank last week. Since there are people who don't drink coffee, it does make sense to have an \(x\)-value of 0. The \(y\)-axis represents the number of times the person went to a shop or restaurant last week to purchase a meal or a drink. It makes no sense to say that a person went -1 times to a shop or restaurant last week to purchase a meal or a drink. Therefore the \(y\)-intercept of this regression line has no practical meaning.
Try It
The scatterplot and regression line below are from a study that collected data from a group of college students on the number of hours per week during the school year they work at a paid job, \(x\), and the number of units they are taking, \(y\). Interpret the \(y\)-intercept of the regression line or explain why it has no practical meaning.
- Answer
-
A student who works 0 hours at a paid job (does not work) is predicted to be taking about 17 units.
Additional video resources: