13: Fundamental Equilibrium Concepts
In this chapter, you will learn how to predict the position of the balance and the yield of a product of a reaction under specific conditions, how to change a reaction's conditions to increase or reduce yield, and how to evaluate an equilibrium system's reaction to disturbances.
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- 13.0: Prelude to Equilibrium
- Reactions can occur in both directions simultaneously (reactants to products and products to reactants) and eventually reach a state of balance. Reactions can occur in both directions simultaneously (reactants to products and products to reactants) and eventually reach a state of balance.
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- 13.1: Chemical Equilibria
- A reaction is at equilibrium when the amounts of reactants or products no longer change. Chemical equilibrium is a dynamic process, meaning the rate of formation of products by the forward reaction is equal to the rate at which the products re-form reactants by the reverse reaction.
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- 13.2: Equilibrium Constants
- For any reaction that is at equilibrium, the reaction quotient Q is equal to the equilibrium constant K for the reaction. If a reactant or product is a pure solid, a pure liquid, or the solvent in a dilute solution, the concentration of this component does not appear in the expression for the equilibrium constant. At equilibrium, the values of the concentrations of the reactants and products are constant and the reaction quotient will always equal K.
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- 13.3: Shifting Equilibria - Le Châtelier’s Principle
- Systems at equilibrium can be disturbed by changes to temperature, concentration, and, in some cases, volume and pressure; volume and pressure changes will disturb equilibrium if the number of moles of gas is different on the reactant and product sides of the reaction. The system's response to these disturbances is described by Le Châtelier's principle: The system will respond in a way that counteracts the disturbance. Not all changes to the system result in a disturbance of the equilibrium.
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- 13.4: Equilibrium Calculations
- The ratios of the rate of change in concentrations of a reaction are equal to the ratios of the coefficients in the balanced chemical equation. The sign of the coefficient of X is positive when the concentration increases and negative when it decreases. We learned to approach three basic types of equilibrium problems. When given the concentrations of the reactants and products at equilibrium, we can solve for the equilibrium constant.
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- 13.E: Fundamental Equilibrium Concepts (Exercises)
- These are homework exercises to accompany the Textmap created for "Chemistry" by OpenStax. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.
Contributors
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


9.3: Questions to Guide Your Reading
1. Describe the environments of Africa.
2. Which environmental challenges and diseases have historically limited population growth in Africa?
3. Discuss the methodological challenges of studying Ancient and Medieval African History.
4. Identify commonly used terms that are potentially problematic for scholars of Africa. Explain why these terms are potentially problematic.
5. What was distinctive about the agriculture of the Ethiopian Highlands?
6. Explain the legend of Queen Makeda and King Solomon and why it remains significant for Ethiopians.
7. How were the states in this region shaped by trade and inter-cultural relationships?
8. Describe the spread of Christianity into Aksum.
9. What characteristics were shared by the Western Sudanic States?
10.
11. Describe the relationship between the Western Sudanic States and the Islamic World.
12. How did nineteenth century European scholars depict the Bantu Migration? What factors influenced their view?
13. How have post-colonial scholars challenged nineteenth century depictions of the Bantu Migration?


2.4: Complex Numbers
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Learning Objectives
- Add and subtract complex numbers.
- Multiply and divide complex numbers.
- Solve quadratic equations with complex numbers
Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.
In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.
Expressing Square Roots of Negative Numbers as Multiples of \(i\)
We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number.The imaginary number \(i\) is defined as the square root of \(−1\).
\[\sqrt{-1}=i\]
So, using properties of radicals,
\[i^2=(\sqrt{-1})^2=-1\]
We can write the square root of any negative number as a multiple of \(i\). Consider the square root of \(−49\).
\[\begin{align*} \sqrt{-49}&= \sqrt{49\times(-1)}\\[4pt] &= \sqrt{49}\sqrt{-1}\\[4pt] &= 7i \end{align*}\]
We use \(7i\) and not \(−7i\) because the principal root of \(49\) is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written \(a+bi\) where \(a\) is the real part and \(b\) is the imaginary part. For example, \(5+2i\) is a complex number. So, too, is \(3+4i\sqrt{3}\).
Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.
Definition: IMAGINARY AND COMPLEX NUMBERS
A complex number is a number of the form \(a+bi\) where
- \(a\) is the real part of the complex number.
- \(b\) is the imaginary part of the complex number.
If \(b=0\), then \(a+bi\) is a real number. If \(a=0\) and \(b\) is not equal to \(0\), the complex number is called a pure imaginary number. An imaginary number is an even root of a negative number.
Howto: Given an imaginary number, express it in the standard form of a complex number
- Write \(\sqrt{-a}\) as \(\sqrt{a}\sqrt{-1}\).
- Express \(\sqrt{-1}\) as \(i\).
- Write \(\sqrt{a}\times i\) in simplest form.
Example \(\PageIndex{1}\): Expressing an Imaginary Number in Standard Form
Express \(\sqrt{-9}\) in standard form.
Solution
\[\begin{align*} \sqrt{-9}&= \sqrt{9}\sqrt{-1)}\\[4pt] &= 3i\\[4pt] \end{align*}\]
In standard form, this is \(0+3i\).
Exercise \(\PageIndex{1}\)
Express \(\sqrt{-24}\) in standard form.
- Answer
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\(\sqrt{-24}=0+2i\sqrt{6}\)
Plotting a Complex Number on the Complex Plane
We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane , which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs \((a,b)\), where \(a\) represents the coordinate for the horizontal axis and \(b\) represents the coordinate for the vertical axis.
Let’s consider the number \(−2+3i\). The real part of the complex number is \(−2\) and the imaginary part is \(3\). We plot the ordered pair \((−2,3)\) to represent the complex number \(−2+3i\), as shown in Figure \(\PageIndex{2}\).
COMPLEX PLANE
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in Figure \(\PageIndex{3}\).
Howto: Given a complex number, represent its components on the complex plane
- Determine the real part and the imaginary part of the complex number.
- Move along the horizontal axis to show the real part of the number.
- Move parallel to the vertical axis to show the imaginary part of the number.
- Plot the point.
Example \(\PageIndex{2}\): Plotting a Complex Number on the Complex Plane
Plot the complex number \(3−4i\) on the complex plane.
Solution
The real part of the complex number is \(3\), and the imaginary part is \(–4\). We plot the ordered pair \((3,−4)\) as shown in Figure \(\PageIndex{4}\).
Exercise \(\PageIndex{2}\)
Plot the complex number \(−4−i\) on the complex plane.
- Answer
Adding and Subtracting Complex Numbers
Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.
COMPLEX NUMBERS: ADDITION AND SUBTRACTION
Adding complex numbers:
\[(a+bi)+(c+di)=(a+c)+(b+d)i\]
Subtracting complex numbers:
\[(a+bi)−(c+di)=(a−c)+(b−d)i\]
Howto: Given two complex numbers, find the sum or difference
- Identify the real and imaginary parts of each number.
- Add or subtract the real parts.
- Add or subtract the imaginary parts.
Example \(\PageIndex{3}\): Adding and Subtracting Complex Numbers
Add or subtract as indicated.
- \((3−4i)+(2+5i)\)
- \((−5+7i)−(−11+2i)\)
Solution
- \[\begin{align*} (3-4i)+(2+5i)&= 3-4i+2+5i\\[4pt] &= 3+2+(-4i)+5i\\[4pt] &= (3+2)+(-4+5)i\\[4pt] &= 5+i \end{align*}\]
- \[\begin{align*} (-5+7i)-(-11+2i)&= -5+7i+11-2i\\[4pt] &= -5+11+7i-2i\\[4pt] &= (-5+11)+(7-2)i\\[4pt] &= 6+5i \end{align*}\]
Exercise \(\PageIndex{3}\)
Subtract \(2+5i\) from \(3–4i\).
- Answer
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\((3−4i)−(2+5i)=1−9i\)
Multiplying Complex Numbers
Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.
Multiplying a Complex Number by a Real Number
Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example, \(3(6+2i)\) :
Howto: Given a complex number and a real number, multiply to find the product
- Use the distributive property.
- Simplify.
Example \(\PageIndex{4}\): Multiplying a Complex Number by a Real Number
Find the product \(4(2+5i)\).
Solution
Distribute the \(4\).
\[\begin{align*} 4(2+5i)&= (4\cdot 2)+(4\cdot 5i)\\[4pt] &= 8+20i \end{align*}\]
Exercise \(\PageIndex{4}\)
Find the product: \(\dfrac{1}{2}(5−2i)\).
- Answer
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\(\dfrac{5}{2}-i\)
Multiplying Complex Numbers Together
Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term, \(i^2\), it equals \(-1\).
\[\begin{align*} (a+bi)(c+di)&= ac+adi+bci+bdi^2\\[4pt] &= ac+adi+bci-bd(-1)\qquad i^2 = -1\\[4pt] &= ac+adi+bci-bd\\[4pt] &= (ac-bd)+(ad+bc)i \end{align*}\]
Group real terms and imaginary terms.
Howto: Given two complex numbers, multiply to find the product
- Use the distributive property or the FOIL method.
- Remember that \(i^2=-1\).
- Group together the real terms and the imaginary terms
Example \(\PageIndex{5}\): Multiplying a Complex Number by a Complex Number
Multiply \((4+3i)(2−5i)\).
Solution
\[\begin{align*} (4+3i)(2-5i)&= 4(2)-4(5i)+3i(2)-(3i)(5i)\\[4pt] &= 8-20i+6i-15(i^2)\\[4pt] &= (8+15)+(-20+6)i\\[4pt] &= 23-14i \end{align*}\]
Exercise \(\PageIndex{5}\)
Multiply: \((3−4i)(2+3i)\).
- Answer
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\(18+i\)
Dividing Complex Numbers
Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form \(a+bi\). We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of \(a+bi\) is \(a−bi\). For example, the product of \(a+bi\) and \(a−bi\) is
\[\begin{align*} (a+bi)(a-bi)&= a^2-abi+abi-b^2i^2\\[4pt] &= a^2+b^2 \end{align*}\]
The result is a real number.
Note that complex conjugates have an opposite relationship: The complex conjugate of \(a+bi\) is \(a−bi\), and the complex conjugate of \(a−bi\) is \(a+bi\). Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide \(c+di\) by \(a+bi\), where neither \(a\) nor \(b\) equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
\[\begin{align*} \dfrac{(c+di)}{(a+bi)}\cdot \dfrac{(a-bi)}{(a-bi)}&= \dfrac{(c+di)(a-bi)}{(a+bi)(a-bi)}\\[4pt] &= \dfrac{ca-cbi+adi-bdi^2}{a^2-abi+abi-b^2i^2} \qquad \text{Apply the distributive property}\\[4pt] &= \dfrac{ca-cbi+adi-bd(-1)}{a^2-abi+abi-b^2(-1)} \qquad \text{Simplify, remembering that } i^2=-1\\[4pt] &= \dfrac{(ca+bd)+(ad-cb)i}{a^2+b^2} \end{align*}\]
Definition: THE COMPLEX CONJUGATE
The complex conjugate of a complex number \(a+bi\) is \(a−bi\). It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
- When a complex number is multiplied by its complex conjugate, the result is a real number.
- When a complex number is added to its complex conjugate, the result is a real number.
Example \(\PageIndex{6}\): Finding Complex Conjugates
Find the complex conjugate of each number.
- \(2+i\sqrt{5}\)
- \(-\dfrac{1}{2}i\)
Solution
- The number is already in the form \(a+bi\). The complex conjugate is \(a−bi\), or \(2−i\sqrt{5}\).
- We can rewrite this number in the form \(a+bi\) as \(0−\dfrac{1}{2}i\). The complex conjugate is \(a−bi\), or \(0+\dfrac{1}{2}i\). This can be written simply as \(\dfrac{1}{2}i\).
Analysis
Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by \(i\).
Exercise \(\PageIndex{6}\)
Find the complex conjugate of \(−3+4i\).
- Answer
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\(−3−4i\)
How to: Given two complex numbers, divide one by the other
- Write the division problem as a fraction.
- Determine the complex conjugate of the denominator.
- Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
- Simplify.
Example \(\PageIndex{7}\): Dividing Complex Numbers
Divide \((2+5i)\) by \((4−i)\).
Solution
We begin by writing the problem as a fraction.
\[\dfrac{(2+5i)}{(4−i)} \nonumber \]
Then we multiply the numerator and denominator by the complex conjugate of the denominator.
\[\dfrac{(2+5i)}{(4−i)}⋅\dfrac{(4+i)}{(4+i)} \nonumber \]
To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).
\[\begin{align*} \dfrac{(2+5i)}{(4-i)}\cdot \dfrac{(4+i)}{(4+i)}&= \dfrac{8+2i+20i+5i^2}{16+4i-4i-i^2}\\[4pt] &= \dfrac{8+2i+20i+5(-1)}{16+4i-4i-(-1)}\; i^2=-1 \\[4pt] &= \dfrac{3+22i}{17}\\[4pt] &= \dfrac{3}{17}+\dfrac{22}{17i} \end{align*}\]
Separate real and imaginary parts.
Note that this expresses the quotient in standard form.
Simplifying Powers of \(i\)
The powers of \(i\) are cyclic. Let’s look at what happens when we raise \(i\) to increasing powers.
\[i^1=i \nonumber \] \[i^2=-1 \nonumber \] \[i^3=i^2⋅i=-1⋅i=-i \nonumber \] \[i^4=i^3⋅i=-i⋅i=-i^2=-(-1)=1 \nonumber \] \[i^5=i^4⋅i=1⋅i=i \nonumber \]
We can see that when we get to the fifth power of i , it is equal to the first power. As we continue to multiply \(i\) by increasing powers, we will see a cycle of four. Let’s examine the next four powers of \(i\).
\[i^6=i^5⋅i=i⋅i=i^2=-1 \nonumber \] \[i^7=i^6⋅i=i^2⋅i=i^3=-i \nonumber \] \[i^8=i^7⋅i=i^3⋅i=i^4=1 \nonumber \] \[i^9=i^8⋅i=i^4⋅i=i^5=i \nonumber \]
The cycle is repeated continuously: \(i,−1,−i,1,\) every four powers.
Example \(\PageIndex{8}\): Simplifying Powers of \(i\)
Evaluate: \(i^{35}\).
Solution
Since \(i^4=1\), we can simplify the problem by factoring out as many factors of \(i^4\) as possible. To do so, first determine how many times \(4\) goes into \(35: 35=4⋅8+3\).
\[i^{35}=i^{4⋅8+3}=i^{4⋅8}⋅i^3={(i^4)}^8⋅i^3=i^8⋅i^3=i^3=−i \nonumber \]
Exercise \(\PageIndex{7}\)
Evaluate: \(i^{18}\)
- Answer
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\(−1\)
Q&A
Can we write \(i^{35}\) in other helpful ways?
As we saw in Example \(\PageIndex{8}\), we reduced \(i^{35}\) to \(i^3\) by dividing the exponent by \(4\) and using the remainder to find the simplified form. But perhaps another factorization of \(i^{35}\) may be more useful. Table \(\PageIndex{1}\) shows some other possible factorizations.
| Factorization of \(i^{35}\) | \(i^{34}⋅i\) | \(i^{33}⋅i^2\) | \(i^{31}⋅i^4\) | \(i^{19}⋅i^{16}\) |
|---|---|---|---|---|
| Reduced form | \({(i^2)}^{17}⋅i\) | \(i^{33}⋅(−1)\) | \(i^{31}⋅1\) | \(i^{19}⋅{(i^4)}^4\) |
| Simplified form | \({(−1)}^{17}⋅i\) | \(−i^{33}\) | \(i^{31}\) | \(i^{19}\) |
Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.
Media
Access these online resources for additional instruction and practice with complex numbers.
- Adding and Subtracting Complex Numbers
- Multiply Complex Numbers
- Multiplying Complex Conjugates
- Raising i to Powers
Key Concepts
- The square root of any negative number can be written as a multiple of \(i\). See Example .
- To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See Example .
- Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See Example .
- Complex numbers can be multiplied and divided.
- The powers of i are cyclic, repeating every fourth one. See Example .


3.3D: Intercostal Nerves
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The anterior divisions of the thoracic spinal nerves (T1–T11) are called the intercostal nerves.
Learning Objectives
- Describe the function of intercostal nerves
Key Points
- The intercostal nerves are part of the somatic nervous system. This enables them to control the contraction of muscles, as well as provide specific sensory information regarding the skin and parietal pleura.
- Intercostal nerves connect to the appropriate ganglion in the sympathetic trunk through rami communicantes and serve the thoracic pleura and the abdominal peritoneum.
- Unlike most other anterior divisions of spinal nerves, the intercostal nerves do not form a plexus.
Key Terms
- thoracic spinal nerves : The spinal nerves emerging from the thoracic vertebrae. Branches also exit the spine and go directly to the sympathetic chain ganglia of the autonomic nervous system where they are involved in the functions of organs and glands in the head, neck, thorax, and abdomen.
- sympathetic trunk : Also called the sympathetic chain or gangliated cord, these are a paired bundle of nerve fibers that run from the base of the skull to the coccyx.
- abdominal peritoneum : The serous membrane that forms the lining of the abdominal cavity. It covers most of the intra-abdominal organs. It is composed of a layer of mesothelium supported by a thin layer of connective tissue. The peritoneum supports the abdominal organs and serves as a conduit for their blood and lymph vessels and nerves.
The intercostal nerves are part of the somatic nervous system and arise from anterior divisions (rami anteriores, ventral divisions) of the thoracic spinal nerves T1 to T11. The intercostal nerves are distributed chiefly to the thoracic pleura and abdominal peritoneum.
Intercostal nerves : An image of the intercostal brachial nerves.
They differ from the anterior divisions of the other spinal nerves in that each pursues an independent course without plexus formation.
First Thoracic Nerve
The anterior division of the first thoracic nerve divides into two branches:
- The larger branch leaves the thorax in front of the neck of the first rib and enters the brachial plexus.
- The other smaller branch, the first intercostal nerve, runs along the first intercostal space and ends on the front of the chest as the first anterior cutaneous branch of the thorax.
The Upper Thoracic Nerves (2nd–6th)
These are limited in their distribution to the parietes (wall) of the thorax. The anterior divisions of the second, third, fourth, fifth, and sixth thoracic nerves, and the small branch from the first thoracic, are confined to the walls of the thorax and are named thoracic intercostal nerves.
Near the sternum, they cross in front of the internal mammary artery and transversus thoracis muscle, pierce the intercostales interni, the anterior intercostal membranes, and pectoralis major, and supply the integument of the front of the thorax and over the mamma, forming the anterior cutaneous branches of the thorax.
The branch from the second nerve unites with the anterior supraclavicular nerves of the cervical plexus.
The Lower Thoracic Nerves (7th–12th)
The seventh intercostal nerve terminates at the xyphoid process, at the lower end of the sternum.
The anterior divisions of the seventh, eighth, ninth, tenth, and eleventh thoracic intercostal nerves are continued anteriorly from the intercostal spaces into the abdominal wall; hence they are named thoraco-abdominal nerves or thoracicoabdominal intercostal nerves.
The tenth intercostal nerve terminates at the umbilicus.
The twelfth (subcostal) thoracic nerve is distributed to the abdominal wall and groin.
Unlike the nerves from the autonomic nervous system that innervate the visceral pleura of the thoracic cavity, the intercostal nerves arise from the somatic nervous system. This enables them to control the contraction of muscles, as well as provide specific sensory information regarding the skin and parietal pleura.
This explains why damage to the internal wall of the thoracic cavity can be felt as a sharp pain localized in the injured region. Damage to the visceral pleura is experienced as an unlocalized ache.


4.13: Summary
This chapter continued where Chapter 3 left off. The focus was still primarily on introducing basic R concepts, but this time at least you can see how those concepts are related to data analysis:
- Installing, loading and updating packages . Knowing how to extend the functionality of R by installing and using packages is critical to becoming an effective R user (Section 4.2 )
- Getting around . Section 4.3 talked about how to manage your workspace and how to keep it tidy. Similarly, Section 4.4 talked about how to get R to interact with the rest of the file system.
- Loading and saving data . Finally, we encountered actual data files. Loading and saving data is obviously a crucial skill, one we discussed in Section 4.5 .
- Useful things to know about variables . In particular, we talked about special values, element names and classes (Section 4.6 ).
- More complex types of variables . R has a number of important variable types that will be useful when analysing real data. I talked about factors in Section 4.7 , data frames in Section 4.8 , lists in Section 4.9 and formulas in Section 4.10 .
- Generic functions . How is it that some function seem to be able to do lots of different things? Section 4.11 tells you how.
- Getting help . Assuming that you’re not looking for counselling, Section 4.12 covers several possibilities. If you are looking for counselling, well, this book really can’t help you there. Sorry.


25.1: Many-Electron Wave Functions
When we have more than one electron, the sixth postulate that we discussed in chapter 24 comes into place. In other words, we need to account for the spin of the electrons and we need the wave function to be antisymmetric with respect to exchange of the coordinates of any two electrons. In order to do so, we can define a new variable \({\bf x}\) which represents the set of all four coordinates associated with an electron: three spatial coordinates \({\bf r}\), and one spin coordinate \(\mathbf{s}\), i.e., \({\bf x} = \{ {\bf r}, {\bf s} \}\). We can then write the electronic wave function as \(\Psi({\bf x}_1, {\bf x}_2, \ldots, {\bf x}_N)\), and we require the sixth postulate to hold by writing:
\[ \Psi\left({\bf x}_1,{\bf x}_2,\ldots, {\bf x}_N\right) = - \Psi\left({\bf x}_2,{\bf x}_1,\ldots, {\bf x}_N\right) \label{26.1.1} \]
A very important step in simplifying \(\Psi({\bf x})\) is to expand it in terms of a set of one-electron functions. Since we need to take into account the spin coordinate as well, we can define a new function, called spin-orbital , by multiplying a spatial orbital by one of the two spin functions:
\[ \begin{equation} \begin{aligned} \chi({\bf x}) &= \psi({\bf r}) \phi_{\uparrow}({\bf s}), \\ \chi({\bf x}) &= \psi({\bf r}) \phi_{\downarrow}({\bf s}). \end{aligned} \end{equation} \label{26.1.2} \]
Notice that for a given spatial orbital \(\psi({\bf r})\), we can form two spin orbitals, one with \(\uparrow\) spin, and one with \(\downarrow\) spin (since the spin coordinate \({\bf s}\) has only two possible values, as already discussed in chapter 23 ). For the spatial orbitals we can use the same one-particle functions that solve the TISEq for the hydrogen atom, \(\psi_{n\ell m_{\ell}}({\bf r})\)(eq. 21.7 in chapter 21 ). Notice how each spin-orbital now depends on four quantum numbers, the three for the spatial part, \(n,\ell,m_{\ell}\), plus the spin quantum number \(m_s\). We need to keep in mind, however, that the spin-orbitals, \(\chi_{n\ell m_{\ell} m_{s}}\), are not analytic solutions to the TISEq, so the resulting wave function is not the exact wave function of the system, but just an approximation.
Once we have defined one-electron spin-orbitals for each electron in the system, we can use them as the basis for our many-electron wave function. While doing so, we need to make sure to enforce the antisymmetry property of the overall wave function. We will start from the simplest case of an atom with two electrons with coordinates \(\mathbf{x}_1\) and \(\mathbf{x}_2\), which we put in two spin-orbitals \(\chi_1\) and \(\chi_2\). We can write the total wave function as a linear combination of the two spin-orbitals as:
\[ \begin{equation} \begin{aligned} \Psi({\bf x}_1, {\bf x}_2) =& b_{11} \chi_1({\bf x}_1) \chi_1({\bf x}_2) + b_{12} \chi_1({\bf x}_1) \chi_2({\bf x}_2) + \\ & b_{21} \chi_2({\bf x}_1) \chi_1({\bf x}_2) + b_{22} \chi_2({\bf x}_1) \chi_2({\bf x}_2). \end{aligned} \end{equation} \label{26.1.3} \]
We then notice that in order for the antisymmetry principle to be obeyed, we need \(b_{12} = -b_{21}\) and \(b_{11} = b_{22} = 0\), which give:
\[ \Psi({\bf x}_1, {\bf x}_2) = b_{12} \left[ \chi_1({\bf x}_1) \chi_2({\bf x}_2) - \chi_2({\bf x}_1) \chi_1({\bf x}_2)\right]. \label{26.1.4} \]
This wave function is sufficient to describe two-electron atoms and ions, such as helium. The numerical coefficient can be determined imposing the normalization condition, and is equal to \(b_{12} = \dfrac{1}{\sqrt{2}}\). For the ground state of helium, we can replace the spatial component of each spin-orbital with the \(1s\) hydrogenic orbital, \(\psi_{100}\), resulting in:
\[ \begin{equation} \begin{aligned} \Psi({\bf x}_1, {\bf x}_2) &= \dfrac{1}{\sqrt{2}} \left[ \psi_{100}({\bf r}_1)\phi_{\uparrow} \; \psi_{100}({\bf r}_2)\phi_{\downarrow} - \psi_{100}({\bf r}_1)\phi_{\downarrow} \; \psi_{100}({\bf r}_2)\phi_{\uparrow} \right] \\ &= \psi_{100}({\bf r}_1)\psi_{100}({\bf r}_2) \dfrac{1}{\sqrt{2}} \left[ \phi_{\uparrow}\phi_{\downarrow} - \phi_{\downarrow}\phi_{\uparrow} \right], \end{aligned} \end{equation} \label{26.1.5} \]
which clearly shows how we need just one spatial orbital, \(\psi_{100}\), to describe the system, while the antisymmetry is taken care by a suitable combination of spin functions, \(\dfrac{1}{\sqrt{2}} \left[ \phi_{\uparrow}\phi_{\downarrow} - \phi_{\downarrow}\phi_{\uparrow} \right]\). Notice also that we commit a small inaccuracy when we say: “two electron occupies one spin-orbital, one electron has spin up, and the other electron has spin down, with configuration: \([\uparrow\downarrow]\)”, as is typically found in general chemistry textbooks. The reality of the spin configuration is indeed more complicated, and the ground state of helium should be represented as \(\dfrac{1}{\sqrt{2}}\left[\uparrow\downarrow-\downarrow\uparrow\right]\).
In order to generalize from two electrons to \(N\), we can first observe how Equation (26.1.4) could be easily constructed by placing the spin-orbitals into a \(2\times2\) matrix and calculating its determinant:
\[ \Psi({\bf x}_1, {\bf x}_2)= \dfrac{1}{\sqrt{2}}{\begin{vmatrix} \chi_1({\bf x}_1)&\chi_2({\bf x}_1)\\\chi_1({\bf x}_2)&\chi_2({\bf x}_2) \end{vmatrix}}, \label{26.1.6} \]
where each column contains one spin-orbital, each row contains the coordinates of a single electron, and the vertical bars around the matrix mean that we need to calculate its determinant. This notation is called the Slater determinant , and it is the preferred way of building any \(N\)-electron wave function. Slater determinants are useful because they can be easily bult for any case of \(N\) electrons in \(N\) spin-orbitals, and they also automatically enforce the antisymmetry of the resulting wave function. A general Slater determinant is written:
\[ \Psi (\mathbf{x} _{1},\mathbf{x} _{2},\ldots ,\mathbf{x} _{N})={\dfrac {1}{\sqrt {N!}}}{\begin{vmatrix}\chi _{1}(\mathbf{x} _{1})&\chi _{2}(\mathbf{x} _{1})&\cdots &\chi _{N}(\mathbf{x} _{1})\\\chi _{1}(\mathbf{x} _{2})&\chi _{2}(\mathbf{x} _{2})&\cdots &\chi _{N}(\mathbf{x} _{2})\\\vdots &\vdots &\ddots &\vdots \\\chi _{1}(\mathbf{x} _{N})&\chi _{2}(\mathbf{x} _{N})&\cdots &\chi _{N}(\mathbf{x} _{N})\end{vmatrix}} = |\chi _{1},\chi _{2},\cdots ,\chi _{N}\rangle, \label{26.1.7} \]
where the notation \(|\cdots\rangle\) is a shorthand to indicate the Slater determinant where only the diagonal elements are reported.


4.2.5J: The Incest Taboo, Marriage, and the Family
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An incest taboo is any cultural rule or norm that prohibits sexual relations between relatives.
Learning Objectives
- Analyze the different constructs of the incest taboo, ranging from biological (the Westermarck effect) to cultural (endogamy and exogamy)
Key Points
- Incest taboo is a cultural norm or rule that forbids sexual relations between relatives.
- Inbreeding is reproduction resulting from the mating of two genetically-related individuals.
- The Westermarck effect is essentially a psychological phenomenon that serves to discourage inbreeding. Through this effect, people who have grown up together are less likely to feel sexually attracted to one another later in life.
- Exogamy is a social arrangement in which marriage is permitted only with members from outside the social group.
- Endogamy is a social arrangement in which marriage can occur only within the same social group.
Key Terms
- exogamy : Marriage to a person belonging to a tribe or group other than your own as required by custom or law.
- inbreeding : Breeding between members of a relatively small population, especially one in which most members are related.
- endogamy : The practice of marrying or being required to marry within one’s own ethnic, religious, or social group.
Inbreeding : An intensive form of inbreeding where an individual S is mated to his daughter D1, granddaughter D2 and so on, in order to maximise the percentage of S’s genes in the offspring. D3 would have 87.5% of his genes, while D4 would have 93.75%.
An incest taboo is any cultural rule or norm that prohibits sexual relations between relatives. All human cultures have norms regarding who is considered suitable and unsuitable as sexual or marriage partners. Usually certain close relatives are excluded from being possible partners. Little agreement exists among cultures about which types of blood relations are permissible partners and which are not. In many cultures, certain types of cousin relations are preferred as sexual and marital partners, whereas others are taboo.
One potential explanation for the incest taboo sees it as a cultural implementation of a biologically evolved preference for sexual partners without shared genes, as inbreeding may have detrimental outcomes.  The existence of the Westermarck effect has achieved some empirical support. The Westermarck effect, first proposed by Edvard Westermarck in 1891, is the theory that children reared together, regardless of biological relationship, form a sentimental attachment that is by its nature non-erotic.
Another school argues that the incest prohibition is a cultural construct that arises as a side effect of a general human preference for group exogamy. Intermarriage between groups construct valuable alliances that improve the ability for both groups to thrive. According to this view, the incest taboo is not necessarily a universal, but it is likely to arise and become stricter under cultural circumstances that favor exogamy over endogamy; it likely to become more lax under circumstances that favor endogamy. This hypothesis has also achieved some empirical support.
Societies that are stratified often prescribe different degrees of endogamy. Endogamy is the opposite of exogamy; it refers to the practice of marriage between members of the same social group. A classic example is seen in India’s caste system, in which unequal castes are endogamous. Inequality between ethnic groups and races also correlates with endogamy. Class, caste, ethnic and racial endogamy typically coexists with family exogamy and prohibitions against incest.


5.8: Naming Molecular Compounds
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Learning Objectives
- Determine the name of a simple molecular compound from its chemical formula.
Molecular Compounds
Molecular compounds are inorganic compounds that take the form of discrete molecules . Examples include such familiar substances as water \(\left( \ce{H_2O} \right)\) and carbon dioxide \(\left( \ce{CO_2} \right)\). These compounds are very different from ionic compounds like sodium chloride \(\left( \ce{NaCl} \right)\). Ionic compounds are formed when metal atoms lose one or more of their electrons to nonmetal atoms. The resulting cations and anions are electrostatically attracted to each other.
So what holds the atoms of a molecule together? Rather than forming ions, the atoms of a molecule share their electrons in such a way that a bond forms between a pair of atoms. In a carbon dioxide molecule, there are two of these bonds, each occurring between the carbon atom and one of the two oxygen atoms.
Larger molecules can have many, many bonds that serve to keep the molecule together. In a large sample of a given molecular compound, all of the individual molecules are identical.
Naming Binary Molecular Compounds
Recall that a molecular formula shows the number of atoms of each element that a molecule contains. A molecule of water contains two hydrogen atoms and one oxygen atom, so its formula is \(\ce{H_2O}\). A molecule of octane, which is a component of gasoline, contains 8 atoms of carbon and 18 atoms of hydrogen. The molecular formula of octane is \(\ce{C_8H_{18}}\).
Naming binary (two-element) molecular compounds is similar to naming simple ionic compounds. The first element in the formula is simply listed using the name of the element. The second element is named by taking the stem of the element name and adding the suffix - ide . A system of numerical prefixes is used to specify the number of atoms in a molecule. Table \(\PageIndex{1}\) lists these numerical prefixes.
| Number of Atoms in Compound | Prefix on the Name of the Element |
|---|---|
| 1 | mono-* |
| 2 | di- |
| 3 | tri- |
| 4 | tetra- |
| 5 | penta- |
| 6 | hexa- |
| 7 | hepta- |
| 8 | octa- |
| 9 | nona- |
| 10 | deca- |
*This prefix is not used for the first element’s name.
Note
- Generally, the less electronegative element is written first in the formula, though there are a few exceptions. Carbon is always first in a formula and hydrogen is after nitrogen in a formula such as \(\ce{NH_3}\). The order of common nonmetals in binary compound formulas is \(\ce{C}\), \(\ce{P}\), \(\ce{N}\), \(\ce{H}\), \(\ce{S}\), \(\ce{I}\), \(\ce{Br}\), \(\ce{Cl}\), \(\ce{O}\), \(\ce{F}\).
- The a or o at the end of a prefix is usually dropped from the name when the name of the element begins with a vowel. As an example, four oxygen atoms, is tetr o xide instead of tetr ao xide.
- The prefix is "mono" is not added to the first element’s name if there is only one atom of the first element in a molecule.
Some examples of molecular compounds are listed in Table \(\PageIndex{2}\).
| Formula | Name |
|---|---|
| \(\ce{NO}\) | nitrogen monoxide |
| \(\ce{N_2O}\) | dinitrogen monoxide |
| \(\ce{S_2Cl_2}\) | disulfur dichloride |
| \(\ce{Cl_2O_7}\) | dichlorine heptoxide |
Notice that the mono- prefix is not used with the nitrogen in the first compound, but is used with the oxygen in both of the first two examples. The \(\ce{S_2Cl_2}\) emphasizes that the formulas for molecular compounds are not reduced to their lowest ratios. The o of the mono- and the a of hepta- are dropped from the name when paired with oxide.
Exercise \(\PageIndex{1}\)
Write the name for each compound.
- CF 4
- SeCl 2
- SO 3
- Answer a:
- carbon tetrafluoride
- Answer b:
- selenium dichloride
- Answer c:
- sulfur trioxide
Simple molecular compounds with common names
For some simple covalent compounds, we use common names rather than systematic names. We have already encountered these compounds, but we list them here explicitly:
- H 2 O: water
- NH 3 : ammonia
- CH 4 : methane
- H 2 O 2 : hydrogen peroxide
Methane is the simplest organic compound. Organic compounds are compounds with carbon atoms and are named by a separate nomenclature system.
Some Compounds Have Both Covalent and Ionic Bonds
If you recall the introduction of polyatomic ions, you will remember that the bonds that hold the polyatomic ions together are covalent bonds. Once the polyatomic ion is constructed with covalent bonds, it reacts with other substances as an ion. The bond between a polyatomic ion and another ion will be ionic. An example of this type of situation is in the compound sodium nitrate. Sodium nitrate is composed of a sodium ion and a nitrate ion. The nitrate ion is held together by covalent bonds and the nitrate ion is attached to the sodium ion by an ionic bond.
Summary
- A molecular compound is usually composed of two or more nonmetal elements.
- Molecular compounds are named with the first element first and then the second element by using the stem of the element name plus the suffix -ide. Numerical prefixes are used to specify the number of atoms in a molecule.


9.2: Intentional Injuries- Violence
The Social-Ecological Model: A Framework for Prevention
The ultimate goal is to stop violence before it begins. Prevention requires understanding the factors that influence violence. CDC uses a four-level social-ecological model to better understand violence and the effect of potential prevention strategies. This model considers the complex interplay between individual, relationship, community, and societal factors. It allows us to understand the range of factors that put people at risk for violence or protect them from experiencing or perpetrating violence. The overlapping rings in the model illustrate how factors at one level influence factors at another level.
Besides helping to clarify these factors, the model also suggests that in order to prevent violence, it is necessary to act across multiple levels of the model at the same time. This approach is more likely to sustain prevention efforts over time than any single intervention.
Individual
The first level identifies biological and personal history factors that increase the likelihood of becoming a victim or perpetrator of violence. Some of these factors are age, education, income, substance use, or history of abuse. Prevention strategies at this level are often designed to promote attitudes, beliefs, and behaviors that ultimately prevent violence. Specific approaches may include education and life skills training.
Relationship
The second level examines close relationships that may increase the risk of experiencing violence as a victim or perpetrator. A person’s closest social circle-peers, partners and family members-influences their behavior and contributes to their range of experience. Prevention strategies at this level may include parenting or family-focused prevention programs, and mentoring and peer programs designed to reduce conflict, foster problem solving skills, and promote healthy relationships.
Community
The third level explores the settings, such as schools, workplaces, and neighborhoods, in which social relationships occur and seeks to identify the characteristics of these settings that are associated with becoming victims or perpetrators of violence. Prevention strategies at this level are typically designed to impact the social and physical environment – for example, by reducing social isolation, improving economic and housing opportunities in neighborhoods, as well as the climate, processes, and policies within school and workplace settings.
Societal
The fourth level looks at the broad societal factors that help create a climate in which violence is encouraged or inhibited. These factors include social and cultural norms that support violence as an acceptable way to resolve conflicts. Other large societal factors include the health, economic, educational and social policies that help to maintain economic or social inequalities between groups in society.
What are Social Norms?
Social norms refer to values, beliefs, attitudes, and/or behaviors shared by a group of people. They are often based on what people believe to be normal, typical, or appropriate. Social norms can function as unspoken rules or guidelines for how people behave, and for how people are expected to behave. People generally follow social norms because they want to fit in with the people around them.
Social norms can result in positive or negative outcomes. Sometimes social norms help people behave in ways that keep themselves – and others – safe and healthy.
Other times, social norms can have the opposite effect, and can lead people to behave in harmful ways.
Social Norms and Violence
Social norms can affect nearly any aspect of our lives. They contribute to our clothing choices, how we speak, our music preferences, and our beliefs about certain social issues. They can also affect our attitudes, beliefs, and behaviors related to violence.
The way we react to violence may be based on what we see other people do, or how we think other people would act. In other words, our reactions are based on what we believe is normal or appropriate.
Misperceptions
People often misperceive (or misunderstand) social norms and overestimate the number of people who behave in unhealthy ways or who accept unhealthy behavior. Researchers have studied college students’ misperceptions of social norms related to alcohol use. They’ve found that most college students do not enjoy heavy drinking. However, they tend to think that heavy drinking is the norm among their peers and overdrink to fit in.
Misperceptions can be harmful when a person alters their own beliefs and behaviors based on a false assumption about other people’s beliefs and behaviors.
Contributors and Attributions
Public Domain Content
- Violence Prevention. Authored by: National Center for Injury Prevention and Control, Division of Violence Prevention. Provided by: Centers for Disease Control and Prevention. Located at: www.cdc.gov/violenceprevention/overview/social-ecologicalmodel.html. License: Public Domain: No Known Copyright
- Social Norms. Authored by: Centers for Disease Control and Prevention. Located at: http://vetoviolence.cdc.gov/violence-prevention-basics-social-norms-change . License: Public Domain: No Known Copyright


Section 04: Rules for quantifiers
For proofs in QL, we use all of the basic rules of SL plus four new basic rules: both introduction and elimination rules for each of the quantifiers.
Since all of the derived rules of SL are derived from the basic rules, they will also hold in QL. We will add another derived rule, a replacement rule called quantifier negation.
Substitution instances
In order to concisely state the rules for the quantifiers, we need a way to mark the relation between quantified sentences and their instances. For example, the sentence \(Pa\) is a particular instance of the general claim ∀\(xPx\).
For a wff \(\mathcal{A}\), a constant \(\mathcal{c}\), and a variable \(\mathcal{x}\), define a substitution instance of ∀\(\mathcal{xA}\) or ∃\(\mathcal{xA}\) is the wff that we get by replacing every occurrence of \(\mathcal{x}\) in \(\mathcal{A}\) with \(\mathcal{c}\). We call \(\mathcal{c}\) the instantiating constant .
To underscore the fact that the variable \(\mathcal{x}\) is replaced by the instantiating constant \(\mathcal{c}\), we will write the original quantified expressions as∀\(\mathcal{x}\)\(\mathcal{Ax}\) and∃\(\mathcal{x}\)\(\mathcal{Ax}\). And we will write the substitution instance \(\mathcal{Ac}\).
Note that \(\mathcal{A}\), \(\mathcal{x}\), and \(\mathcal{c}\) are all meta-variables. That is, they are stand-ins for any wff, variable, and constant whatsoever. And when we write \(\mathcal{Ac}\), the constant \(\mathcal{c}\) may occur multiple times in the wff \(\mathcal{A}\).
For example:
~ \(Aa\) → \(Ba\), \(Af\) → \(Bf\), and \(Ak\) → \(Bk\) are all substitution instances of ∀\(x\)(\(Ax\) → \(Bx\)); the instantiating constants are \(a\), \(f\), and \(k\), respectively.
~ \(Raj\), \(Rdj\), and \(Rjj\) are substitution instances of ∃\(zRzj\); the instantiating constants are \(a\), \(d\), and \(j\), respectively.
Universal elimination
If you have ∀\(xAx\), it is legitimate to infer that anything is an \(A\). You can infer \(A\) a , \(A\) b , \(A\) z , \(Ad\) 3 . You can infer any substitution instance, \(A\(\mathcal{c}\) for any constant \(\mathcal{c}\).
This is the general form of the universal elimination rule (∀E):
When using the ∀E rule, you write the substituted sentence with the constant \(\mathcal{c}\) replacing all occurrences of the variable \(\mathcal{x}\) in \(\mathcal{A}\). For example:
Existential introduction
It is legitimate to infer ∃\(xPx\) if you know that something is a \(P\). It might be any particular thing at all. For example, if you have \(Pa\) available in the proof, then ∃\(xPx\) follows.
This is the existential introduction rule (∃I):
It is important to notice that the variable \(\mathcal{x}\) does not need to replace all occurrences of the constant \(\mathcal{c}\). You can decide which occurrences to replace and which to leave in place. For example:
Universal introduction
A universal claim like ∀\(xPx\) would be proven if every substitution instance of it had been proven. That is, if every sentence \(Pa\), \(Pb\), ... were available in a proof, then you would certainly be entitled to claim ∀\(xPx\). Alas, there is no hope of proving every substitution instance. That would require proving \(Pa\), \(Pb\), ..., \(P\)j 2 , ..., \(Ps\) 7 , ..., and so on to infinity. There are infinitely many constants in QL, and so this process would never come to an end.
Consider instead a simple argument: ∀\(xMx\), .˙. ∀\(yMy\)
It makes no difference to the meaning of the sentence whether we use the variable \(x\) or the variable \(y\), so this argument is obviously valid. Suppose we begin in this way:
We have derived \(Ma\). Nothing stops us from using the same justification to derive \(Mb\), ..., \(Mj\) 2 , ..., \(Ms\) 7 , ..., and so on until we run out of space or patience. We have effectively shown the way to prove \(M\)\(\mathcal{c}\) for any constant \(\mathcal{c}\). From this, ∀\(yMy\) follows.
It is important here that \(a\) was just some arbitrary constant. We had not made any special assumptions about it. If \(Ma\) were a premise of the argument, then this would not show anything about all \(y\). For example:
This is the schematic form of the universal introduction rule (∀I):
∗ The constant \(\mathcal{c}\) must not occur in any undischarged assumption.
Note that we can do this for any constant that does not occur in an undischarged assumption and for any variable.
Note also that the constant may not occur in any undischarged assumption, but it may occur as the assumption of a subproof that we have already closed. For example, we can prove ∀\(z\)(\(Dz\) → \(Dz\)) without any premises.
Existential elimination
A sentence with an existential quantifier tells us that there is some member of the UD that satisfies a formula. For example, ∃\(xSx\) tells us (roughly) that there is at least one \(S\). It does not tell us which member of the UD satisfies \(S\), however. We cannot immediately conclude \(Sa\), \(Sf\) 23 , or any other substitution instance of the sentence. What can we do? Suppose that we knew both ∃\(xSx\) and ∀\(x\)(\(Sx\) → \(Tx\)). We could reason in this way:
Since ∃\(xSx\), there is something that is an \(S\). We do not know which constants refer to this thing, if any do, so call this thing ‘Ishmael’. From ∀\(x\)(\(Sx\) → \(Tx\)), it follows that if Ishmael is an \(S\), then it is a \(T\). Therefore, Ishmael is a \(T\). Because Ishmael is a \(T\), we know that ∃\(xTx\).
In this paragraph, we introduced a name for the thing that is an \(S\). We gave it an arbitrary name (‘Ishmael’) so that we could reason about it and derive some consequences from there being an \(S\). Since ‘Ishmael’ is just a bogus name introduced for the purpose of the proof and not a genuine constant, we could not mention it in the conclusion. Yet we could derive a sentence that does not mention Ishmael; namely, ∃\(xTx\). This sentence does follow from the two premises.
We want the existential elimination rule to work in a similar way. Yet since English language words like ‘Ishmael’ are not symbols of QL, we cannot use them in formal proofs. Instead, we will use constants of QL which do not otherwise appear in the proof.
A constant that is used to stand in for whatever it is that satisfies an existential claim is called a proxy . Reasoning with the proxy must all occur inside a subproof, and the proxy cannot be a constant that is doing work elsewhere in the proof.
This is the schematic form of the existential elimination rule (∃E):
∗ The constant \(\mathcal{c}\) must not appear in ∃\(\mathcal{xAx}\), in \(\mathcal{B}\), or in any undischarged assumption.
Since the proxy constant is just a place holder that we use inside the subproof, it cannot be something that we know anything particular about. So it cannot appear in the original sentence ∃\(\mathcal{xAx}\) or in an undischarged assumption. Moreover, we do not learn anything about the proxy constant by using the ∃E rule. So it cannot appear in \(\mathcal{B}\), the sentence you prove using ∃E.
The easiest way to satisfy these requirements is to pick an entirely new constant when you start the subproof, and then not to use that constant anywhere else in the proof. Once you close the subproof, do not mention it again.
With this rule, we can give a formal proof that∃\(xSx\) and∀\(x\)(\(Sx\) → \(Tx\)) together entail ∃x\(Tx\).
Notice that this has effectively the same structure as the English-language argument with which we began, except that the subproof uses the proxy constant ‘\(i\)’ rather than the bogus name ‘Ishmael’.
Quantifier negation
When translating from English to QL, we noted that ¬∃\(x\)¬\(\mathcal{A}\) is logically equivalent to ∀\(x\)\(\mathcal{A}\). In QL, they are provably equivalent. We can prove one half of the equivalence with a rather gruesome proof:
In order to show that the two sentences are genuinely equivalent, we need a second proof that assumes ¬∃\(x\)¬\(\mathcal{A}\) and derives ∀\(x\)\(\mathcal{A}\). We leave that proof as an exercise for the reader.
It will often be useful to translate between quantifiers by adding or subtracting negations in this way, so we add two derived rules for this purpose. These rules are called quantifier negation (QN):
¬∀\(\mathcal{xA}\) ⇐⇒∃\(\mathcal{x}\)¬\(\mathcal{A}\)
¬∃\(\mathcal{xA}\) ⇐⇒∀\(\mathcal{x}\)¬\(\mathcal{A}\) QN
Since QN is a replacement rule, it can be used on whole sentences or on subformulae.


16.3: Nouns
- Identify and use various types of nouns
Nouns are a diverse group of words, and they are very common in English. Nouns refer to things —the names of people (Dr. Sanders, lawyers), places (Kansas, factory, home), things (scissors, sheet music, book), or ideas (love, truth, beauty, intelligence). There are many to categorize nouns. Let’s take a look at each of these classifications and see exactly what they each mean.
Common vs. Proper Nouns
A proper noun refers to a specific person, place, organization, etc. Proper nouns are capitalized because they are specific nouns. Some examples of proper nouns are Steven, Apple (the company), New York, and the Seattle Seahawks.
A common noun refers to a general group or class of people, places, objects, etc. Common nouns are generic words, like tissue or watch . They are always lowercase (unless they begin a sentence).
- common noun: girl
- proper noun: Esther
Note: This rule also applies to adjectives that are based on proper nouns:
- It can be difficult to understand Shakespearian language.
- After her encounter with Lukas, Elisa vowed to hate all Swiss men.
However, if you’re talking about swiss cheese , pasteurized milk , and french fries , these adjectives are lowercase. They have a nonliteral meaning: the cheese isn’t really from Switzerland, Louie Pasteur didn’t treat the milk himself, and the fries aren’t really from France.
Concrete vs. Abstract Nouns
Concrete nouns are things you can touch, see, hear, or otherwise sense, like book , light , or warmth .
Abstract nouns, on the other hand, are (as you might expect) abstract concepts that can’t be perceived through the senses, such as time and love .
- concrete noun: rock
- abstract noun: justice
Collective Nouns
Nouns can get a little tricky when it comes to a discussion of collective nouns. Collective nouns are nouns such as family , team , and majority . The tricky part comes when we have to make a decision about whether these nouns are singular or plural because we have to choose verbs that will agree with these nouns.
And, now, here’s the really tricky part: There are no hard and fast rules. The verb you choose to agree with the collective noun actually depends upon how you want your readers to perceive the noun. Is it a single unit or a group of individuals? Even then, it depends upon context. Take the collective noun family , for example.
- The family have all gone their separate ways since Grandma died.
- The whole family is coming to my house for Thanksgiving this year. I had better learn to cook a turkey.
In her book, Rhetorical Grammar , Martha Kolln (1991) says “[collective nouns] can be treated as either singular or plural, depending on context and meaning” (p. 47). So, it really does depend on the situation.
You may be wondering how this information is helpful. The key is to think about how you might perceive the collective noun and then, of course, to consider how it’s used in the sentence. And, after all, there are only about 200 collective nouns in the English language, so you really only have to worry about 200 of these. Okay, that’s a lot. But this is a great example of how, very often, there are no hard and fast rules for grammar.
Verbal Nouns
A verbal noun is a type of noun that is derived from a verb. It looks like a verb but actually functions in a sentence like a noun. Here are some examples:
Running from zombies is hard work.
Jogging is a good exercise that will help you prepare, but you have to do it every day.
We had a meeting to compare our zombie action plans.
Verbal nouns and something called gerunds (a form of a verb or verb phrase that functions as a noun phrase and subject in a sentence) are very similar. In fact, the first two examples above are examples of verbal nouns that are also gerunds. But, a verbal noun can be more than a gerund. In the last example, the word meeting is functioning like a noun but isn’t a noun phrase that’s the subject of a sentence.
It can certainly get a little confusing, and even the grammar experts disagree sometimes about the differences between verbal nouns and gerunds.
The key thing for you to remember is that, when we are talking about nouns, verbs can sometimes function in your sentences like nouns.
Compound Nouns
A compound noun is a noun that is the result of joining together two other words (such as tooth and paste making toothpaste).
A compound noun acts like one word, despite being a combination of two. Compound nouns can be classified as closed, hyphenated, or open. A closed compound takes the form of two words put together with no space, such as daydream. A hyphenated compound includes two or more words joined by a hyphen, such as dry-cleaning. An open compound is two words separated by a space but acting as one unit, such as vacuum cleaner.
One common misconception is that compounds are hyphenated or open when one of the root words is longer than one syllable. However, it is important to remember that there are many open or hyphenated compound nouns that have of two single-syllable root words, such as six-pack.
Some compound nouns differ in writing style depending on who you ask, while others are recently developed, such as e-mail being shortened further to email.
Types of Compound Nouns
- Solid or Closed form : These compound nouns are defined as being a complete conjoining between the two words that form its makeup. Examples of closed compound nouns include: watermelon, underground, catfish and skydiving.
- Hyphenated form : Compound nouns that often appear in a hyphenated form are nouns that have suffixes (such as fund-rais(er) and wire-fasten(er)) and nouns that contain articles, conjunctions, or prepositions (such as mother-in-law).
- Open or Spaced form : These are compound nouns that are considered to be compound even though they are separated by a space just like any two words. Despite looking like two independent words, they act together as one. Examples of open compound nouns are science fiction, peanut butter, and address book.
Hyphens are often considered a squishy part on language (we’ll discuss this further in Hyphens and Dashes ). Because of this, usage differs and often depends on the individual choice of the writer rather than on a hard-and-fast rule. This means open, hyphenated, and closed forms may be encountered for the same compound noun, such as the triplets container ship/container-ship/containership and particle board/particle-board/particleboard. If you’re ever in doubt whether a compound should be closed, hyphenated, or open, dictionaries are your best reference.
Review the types of nouns in the following exercise.
https://lumenlearning.h5p.com/content/1290907310938413418/embed
Plurals
The process of making compound nouns plural has its own set of conventions to follow. In all forms of compound nouns, we pluralize the chief element of a compound word (i.e., we pluralize the primary noun of the compound).
- fisher man → fisher men
- black bird → black birds
- brother -in-law → brothers -in-law
The word hand-me-down doesn’t have a distinct primary noun, so its plural is hand-me-downs .
Remember that compounds may be written in three different ways: the solid or closed form , the hyphenated form , and the open or spaced form .
https://assessments.lumenlearning.co...essments/20331
Pluralization
English has both regular and irregular plural nouns. Regular plurals follow a rule (and other similar rules) so you generally know how to pluralize them, but irregular plurals are, well, not regular and don’t follow a “standard” rule.
Regular Plurals
Let’s start with regular plurals: regular plural nouns use established patterns to indicate that there is more than one of a thing. As was mentioned earlier, we add the plural suffix – s or – es to most words ( cats , zebras , classes , foxes, heroes ). Remember that when words have a foreign origin (e.g., Latin, Greek, Spanish), we just add the plural suffix – s ( tacos , avocados , maestros ).
When a word ends in y and there is a consonant before y , we change the y to i and add – es . Thus sky becomes skies . However, if the y follows another vowel, you simply add an – s . ( donkeys , alloys ). When a word ends in – f or – fe , we change the f to v and add – es ( calves , leaves ). However, if there are two terminal f s, or if you still pronounce the f in the plural, you simply add an – s ( cliffs , chiefs ).
Irregular Plurals
Irregular plurals , unlike regular plurals, don’t necessarily follow any particular pattern—instead, they follow a lot of different patterns. For this reason, irregular plurals require a lot of memorization. If you’re ever in doubt, the dictionary is there for you.
The first kind of irregular plural we’ll talk about is the no-change or base plural . In these words, the singular noun has the exact same form as the plural ( sheep , fish , deer , moose ). Most no-change plurals are types of animals.
The next type of irregular is the mid-word vowel change . This includes words like tooth , man , and mouse , which become teeth , men , and mice .
Note: The plural for a computer mouse (as opposed to the fuzzy animal) can either be mice or mouses . Some people prefer mouses as it creates some differentiation between the two words.
We also have the plural – en . In these words, – en is used as the plural ending instead of – s or -es .
- child → children
- ox → oxen
- brother → brethren
- sister → sistren
Note: Brethren and sistren are antiquated terms that you’re unlikely to encounter or use; however, since these are the only four words in English that use this plural, all four have been included above.
The last category of irregular plurals is borrowed words . These words are native to other languages (e.g., Latin, Greek) and have retained the pluralization rules from their original tongue.
| Singular – us ; Plural – i | cactus → cacti | fungus → fungi | syllabus → syllabi |
|---|---|---|---|
| Singular – a ; Plural – ae | formula → formulae | vertebra → vertebrae | larva → larvae |
| Singular – ix , – ex ; Plural – ices , – es | appendix → appendices | matrix → matrices | index → indices |
| Singular – on , – um ; Plural – a | bacterium → bacteria | criterion → criteria | medium → media |
| Singular – is ; Plural – es | thesis → theses | analysis → analyses | crisis → crises |
The rules presented in the table above are almost always followed, but as a borrowed word becomes more popular in its usage, it can be adopted into regular pluralization. For example, formulas and appendixes are accepted words in formal situations. Also, in informal speech, cactuses and funguses are acceptable.
Note: Because of the word’s history, octopuses is preferred to octopi , but octopi is an accepted word.
https://lumenlearning.h5p.com/content/1290622187380800738/embed
[reveal-answer q=”245005″]Explanations of the answers[/reveal-answer]
[hidden-answer a=”245005″]
| reefs | reefs is the plural of reef . – s is added. It is an exception to the rule. | boys | boys is the plural of boy . – s is added because the y follows a vowel | waltz | waltz is the singular of waltzes, which is a regular plural noun. |
| memorandum | memorandum is the singular of memoranda . The singular ends with – um , so the plural ends with – a . | hypothesis | hypothesis is the singular of hypotheses . The singular ends with – is , so the plural ends with – es . | phenomena | phenomena is the plural of phenomenon . The singular ends with – on , so the plural ends with – a . |
| focus | focus is the singular of foci or focuses . The singular ends with – us , so the plural typically ends with – i , but – es is also acceptable | vertebrae | vertebrae is the plural of vertebra . The singular ends with – a , so the plural ends with – ae . | appendices | appendices is the plural of appendix . The singular ends with – ix , so the plural ends with – ices . |
| children | children is the plural of child . This is an – en noun. To form the plural, – ren was added. | squid | This is a no-change plural. The singular and plural have the same form, so squid could be singular or plural | man | man is the singular of men . This is a mid-word vowel-change plural. The a in man was changed to an e. |
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Count vs. Non-count Nouns
Count nouns are nouns that can be counted. Count nouns can be associated with a numerical value (i.e., three whales) in both its singular and plural forms (one fox, two foxes). In some cases, the number can be replaced by the words a, an, or the (a fox, an owl, the squirrel).
If a noun cannot have a numerical value nor a plural form, it is called a non-count or mass noun . A person can give another person advice, but they cannot give three advices, because advice has no quantity. Other examples of non-count nouns include nouns like weather, happiness, homework, furniture, thunder, etc. You will know that something is a non-count noun if it sounds weird when you try to pluralize it by adding an -s to the end. For example, you can’t have “furnitures” or “thunders,” but you can have “pieces of furniture” or “claps of thunder.”
The adjectives less and fewer are both used to indicate a smaller amount of the noun they modify. Many and much are used to indicate a large amount of something. People often use these pairs words interchangeably; however, the words fewer and many are used with count nouns, while less and much are used with non-count nouns:
- The pet day care has fewer dogs than cats this week.
- Next time you make these cookies, you should use less sugar.
- Many poets struggle when they try to determine whether a poem is complete or not.
- There’s too much goodness in her heart.
You may have noticed that much has followed the adverb too in this example ( too much ). This is because you rarely find much by itself. You don’t really hear people say things like “Now please leave me alone; I have much research to do.” The phrase “a lot of” has taken its place in current English: “I have a lot of research to do.” A lot of can be used in the place of either many or much :
- A lot of poets struggle when they try to determine whether a poem is finished or not.
- There’s a lot of goodness in her heart.
https://lumenlearning.h5p.com/content/1290627472724984348/embed
[reveal-answer q=”842967″]Explain Answers[/reveal-answer]
[hidden-answer a=”842967″]
-
There was
much
food at the event. There were
fewer
soups than salads and even
fewer
desserts.
- Food is non-count, so it takes much, not many . Soups and desserts are both count, so they take fewer not less .
- Even though much is technically correct, you may want to use a lot instead. It sounds less antiquated.
-
Miguel loved studying
outer space
—especially different
galaxies
.
- Outer space is non-count, so it does not have a plural. Galaxy is count, so it does have a plural. Since we are talking about different items, there must be more than one, so galaxies is correct.
-
Arturo had too much
water
before his workout.
- Much must be followed by a non-count noun. Of the two options ( water and drinks ), water is the non-count noun. If many were used instead of much , the correct sentence would be “Arturo had too many drinks before his workout.”
-
You can only be in this line if you have fifteen items or
fewer
.
- Because items is a count noun, fewer is required here. This may surprise you, since many stores have a “fifteen items or less” line, but using less is grammatically incorrect. However, this phrase has become so common that stores sound uppity if they use fewer instead of less . Some stores get around this issue by saying “around fifteen items” instead.
-
Evelyn was disappointed in the weather forecast;
a lot of
rain was predicted. She preferred dry weather.
- While much would also fit in this blank, the phrase a lot of is much more common and more likely to be used.
-
I had a lengthy list containing
many
ideas for the project.
- The adjective is modifying the count-noun ideas , so many is needed in this example.
[/hidden-answer]
Contributors and Attributions
- Text: Pluralization. Provided by : Lumen Learning. License : CC BY: Attribution
- Common vs. Proper Nouns and Concrete vs. Abstract Nouns. Provided by : Lumen Learning. License : CC BY: Attribution
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY-SA: Attribution-ShareAlike
- Image of two men. Authored by : Gregor Cresnar. Provided by : The Noun Project. Located at : https://thenounproject.com/search/?q=people&i=176708 . License : CC BY: Attribution
- Compound noun. Provided by : Teflpedia. Located at : http://teflpedia.com/Compound_noun . License : CC BY-SA: Attribution-ShareAlike
- B5bugerbear. Authored by : CryptoTom. Located at : https://commons.wikimedia.org/wiki/File:B5bugerbear_Colored_Version.jpg . License : CC BY-SA: Attribution-ShareAlike
- Marina, Large Shoes. Authored by : Dave Parker. Located at : https://www.flickr.com/photos/daveparker/4320683221 . License : CC BY-SA: Attribution-ShareAlike
- Nouns. Authored by : Excelsior OWL. Located at : https://owl.excelsior.edu/grammar-essentials/parts-of-speech/nouns/nouns-try-it-out/ . License : Public Domain: No Known Copyright


Nora . I don't believe that any longer. I believe that before all else I am a reasonable human being, just as you are--or, at all events, that I must try and become one. I know quite well, Torvald, that most people would think you right, and that views of that kind are to be found in books; but I can no longer content myself with what most people say, or with what is found in books. I must think over things for myself and get to understand them.
Helmer . Can you not understand your place in your own home? Have you not a reliable guide in such matters as that?--have you no religion?
Nora . I am afraid, Torvald, I do not exactly know what religion is.
Helmer . What are you saying?
Nora . I know nothing but what the clergyman said, when I went to be confirmed. He told us that religion was this, and that, and the other. When I am away from all this, and am alone, I will look into that matter too. I will see if what the clergyman said is true, or at all events if it is true for me.
Helmer . This is unheard of in a girl of your age! But if religion cannot lead you aright, let me try and awaken your conscience. I suppose you have some moral sense? Or--answer me--am I to think you have none?
Nora . I assure you, Torvald, that is not an easy question to answer. I really don't know. The thing perplexes me altogether. I only know that you and I look at it in quite a different light. I am learning, too, that the law is quite another thing from what I supposed; but I find it impossible to convince myself that the law is right. According to it a woman has no right to spare her old dying father, or to save her husband's life. I can't believe that.
Helmer .  You don't understand the conditions of the world in which you live.
Nora . No, I don't. But now I am going to try. I am going to see if I can make out who is right, the world or I.
Helmer .
Nora . I have never felt my mind so clear and certain as tonight.
Helmer . And is it with a clear and certain mind that you forsake your husband and your children?
Nora . Yes, it is.
Helmer . Then there is only one possible explanation.
Nora . What is that?
Helmer . You do not love me anymore.
Nora . No, that is just it.
Helmer . Nora!--and you can say that?
Nora . It gives me great pain, Torvald, for you have always been so kind to me, but I cannot help it. I do not love you any more.
Helmer [regaining his composure] . Is that a clear and certain conviction too?
Nora . Yes, absolutely clear and certain. That is the reason why I will not stay here any longer.
Helmer . And can you tell me what I have done to forfeit your love?
Nora . Yes, indeed I can. It was tonight, when the wonderful thing did not happen; then I saw you were not the man I had thought you were.
Helmer . Explain yourself better. I don't understand you.
Nora . I have waited so patiently for eight years; for, goodness knows, I knew very well that wonderful things don't happen every day. Then this horrible misfortune came upon me; and then I felt quite certain that the wonderful thing was going to happen at last. When Krogstad's letter was lying out there, never for a moment did I imagine that you would consent to accept this man's conditions. I was so absolutely certain that you would say to him: Publish the thing to the whole world. And when that was done--
Helmer . Yes, what then?--when I had exposed my wife to shame and disgrace?
Nora . When that was done, I was so absolutely certain, you would come forward and take everything upon yourself, and say: I am the guilty one.
Helmer . Nora--!
Nora . You mean that I would never have accepted such a sacrifice on your part? No, of course not. But what would my assurances have been worth against yours? That was the wonderful thing which I hoped for and feared; and it was to prevent that, that I wanted to kill myself.
Helmer . I would gladly work night and day for you, Nora--bear sorrow and want for your sake. But no man would sacrifice his honour for the one he loves.
Nora . It is a thing hundreds of thousands of women have done.
Helmer . Oh, you think and talk like a heedless child.
Nora . Maybe. But you neither think nor talk like the man I could bind myself to. As soon as your fear was over--and it was not fear for what threatened me, but for what might happen to you--when the whole thing was past, as far as you were concerned it was exactly as if nothing at all had happened. Exactly as before, I was your little skylark, your doll, which you would in future treat with doubly gentle care, because it was so brittle and fragile. [Getting up.] Torvald--it was then it dawned upon me that for eight years I had been living here with a strange man, and had borne him three children--. Oh, I can't bear to think of it! I could tear myself into little bits!
Helmer [sadly] . I see, I see. An abyss has opened between us--there is no denying it. But, Nora, would it not be possible to fill it up?
Nora . As I am now, I am no wife for you.
Helmer . I have it in me to become a different man.
Nora . Perhaps--if your doll is taken away from you.
Helmer . But to part!--to part from you! No, no, Nora, I can't understand that idea.
Nora [going out to the right] . That makes it all the more certain that it must be done.
Helmer . Nora, Nora, not now! Wait until tomorrow.
Nora [putting on her cloak] . I cannot spend the night in a strange man's room.
Helmer . But can't we live here like brother and sister--?
Nora [putting on her hat] . You know very well that would not last long. [Puts the shawl round her.] Goodbye, Torvald. I won't see the little ones. I know they are in better hands than mine. As I am now, I can be of no use to them.
Helmer . But some day, Nora--some day?
Nora . How can I tell? I have no idea what is going to become of me.
Helmer . But you are my wife, whatever becomes of you.
Nora . Listen, Torvald. I have heard that when a wife deserts her husband's house, as I am doing now, he is legally freed from all obligations towards her. In any case, I set you free from all your obligations. You are not to feel yourself bound in the slightest way, any more than I shall. There must be perfect freedom on both sides. See, here is your ring back. Give me mine.
Helmer . That too?
Nora . That too.
Helmer . Here it is.
Nora . That's right. Now it is all over. I have put the keys here. The maids know all about everything in the house--better than I do. Tomorrow, after I have left her, Christine will come here and pack up my own things that I brought with me from home. I will have them sent after me.
Helmer . All over! All over!--Nora, shall you never think of me again?
Nora . I know I shall often think of you, the children, and this house.
Helmer . May I write to you, Nora?
Nora . No--never. You must not do that.
Helmer . But at least let me send you--
Nora . Nothing--nothing--
Helmer . Let me help you if you are in want.
Nora . No. I can receive nothing from a stranger.
Helmer . Nora--can I never be anything more than a stranger to you?
Nora [taking her bag] . Ah, Torvald, the most wonderful thing of all would have to happen.
Helmer . Tell me what that would be!
Nora . Both you and I would have to be so changed that--. Oh, Torvald, I don't believe any longer in wonderful things happening.
Helmer . But I will believe in it. Tell me! So changed that--?
Nora . That our life together would be a real wedlock. Goodbye. [She goes out through the hall.]
Helmer [sinks down on a chair at the door and buries his face in his hands] . Nora! Nora! [Looks round, and rises.] Empty. She is gone. [A hope flashes across his mind.] The most wonderful thing of all--?
[The sound of a door shutting is heard from below.]


6.3: Examples of Male Reproductive Terms Easily Defined By Their Word Components
Here are examples of common medical terms related to the male reproductive system that can be easily defined by breaking the terms into their word components.
Prostatitis
- Break down the medical term into word components: Prostat / itis
- Label the word components: Prostat = WR; itis = S
- Define the word components: Prostat = prostate; itis = inflammation
- Create a final definition of the medical term: Inflammation of the prostate gland
Transurethral
- Break down the medical term into word components: Trans / urethr / al
- Label the word components: Trans = P; urethr = WR; al = S
- Define the word components: T rans = through; urethr = urethra; al = pertaining to
- Create a final definition of the medical term: Pertaining to through the urethra
Vasectomy
- Break down the medical term into word components: Vas / ectomy
- Label the word components: Vas = WR; ectomy = S
- Define the word components: Vas = vas deferens duct; ectomy = excision or removal of
- Create a final definition of the medical term: Excision or removal of the vas deferens duct
Interactive Learning Activity: Practice defining and pronouncing male reproductive system medical terms by breaking them down into word parts.
Query \(\PageIndex{1}\)
You can also print this as a Chapter 6 Student Companion Worksheet and check your answers with this Answer Key .