8.4: Other Forms of Energy
Thermal Energy
From all the foregoing, it is clear that when an interaction can be completely described by a potential energy function we can define a quantity, which we have called the total mechanical energy of the system, \(E_{mech} = K + U\), that is constant throughout the interaction. However, we already know from our study of collisions that this is rarely the case. Essential to the concept of potential energy is the idea of “storage and retrieval” of the kinetic energy of the system during the interaction process. When kinetic energy simply disappears from the system and does not come back, a full description of the process in terms of a potential energy is not possible.
Processes in which some amount of mechanical energy disappears (that is, it cannot be found anywhere anymore as either macroscopic kinetic or potential energy) are called dissipative . Mysterious as they may appear at first sight, there is actually a simple, intuitive explanation for them. All macroscopic systems consist of a great number of small parts that enjoy, at the microscopic level, some degree of independence from each other and from the body to which they belong. Macroscopic motion of an object requires all these parts to move together as a whole, at least on average; however, a collision with another object may very well “rattle” all these parts and leave them in a more or less disorganized state. If the total energy is conserved, then after the collision the object’s atoms or molecules may be, on average, vibrating faster or banging against each other more often than before, but they will do so in random directions, so this increased “agitation” will not be perceived as macroscopic motion of the object as a whole.
This kind of random agitation at the microscopic level that I have just introduced is what we know today as thermal energy , and it is by far the most common “sink” or reservoir where macroscopic mechanical energy is “dissipated.” In our example of an inelastic collision, the energy the objects had is not gone from the universe, in fact it is still right there inside the objects themselves; it is just in a disorganized or incoherent state from which, as you can imagine, it would be pretty much impossible to retrieve it, since we would have to somehow get all the randomly-moving parts to get back to moving in the same direction again.
We will have a lot more to say about thermal energy in a later chapter, but for the moment you may want to think of it as essentially noise : it is what is left (the residual motional or configurational energy, at the microscopic level) after you remove the average, macroscopically-observable kinetic or potential energy. So, for example, for a solid object moving with a velocity \(v_{cm}\), the kinetic part of its thermal energy would be the sum of the kinetic energies of all its microscopic parts, calculated in its center of mass (or zero-momentum) reference frame ; that way you remove from every molecule’s velocity the quantity \(v_{cm}\), which they all must have in common—on average (since the body as a whole is moving with that velocity).
In order to establish conservation of energy as a fact (which was one of the greatest scientific triumphs of the 19th century) it was clearly necessary to show experimentally that a certain amount of mechanical energy lost always resulted in the same predictable increase in the system’s thermal energy. Thermal energy is largely “invisible” at the macroscopic level, but we detect it indirectly through an object’s temperature . The crucial experiments to establish what at the time was called the “mechanical equivalent of heat” were carried out by James Prescott Joule in the 1850’s, and required exceedingly precise measurements of temperature (in fact, getting the experiments done was only half the struggle; the other half was getting the scientific establishment to believe that he could measure changes in temperature so accurately!)
Fundamental Interactions
At the most fundamental (microscopic) level, physicists today believe that there are only four (or three, depending on your perspective) basic interactions: gravity, electromagnetism, the strong nuclear interaction (responsible for holding atomic nuclei together), and the weak nuclear interaction (responsible for certain nuclear processes, such as the transmutation of a proton into a neutron 1 and vice-versa). In a technical sense, at the quantum level, electromagnetism and the weak nuclear interactions can be regarded as separate manifestations of a single, consistent quantum field theory, so they are sometimes referred to as “the electroweak interaction.”
All of these interactions are conservative, in the sense that for all of them one can define the equivalent of a “potential energy function” (generalized, as necessary, to conform to the requirements of quantum mechanics and relativity), so that for a system of elementary particles interacting via any one of these interactions the total kinetic plus potential energy is a constant of the motion. For gravity (which we do not really know how to “quantize” anyway!), this function immediately carries over to the macroscopic domain without any changes, as we shall see in a later chapter, and the gravitational potential energy function I introduced earlier in this chapter is an approximation to it valid near the surface of the earth (gravity is such a weak force that the gravitational interaction between any two earth-bound objects is virtually negligible, so we only have to worry about gravitational energy when one of the objects involved is the earth itself).
As for the strong and weak nuclear interactions, they are only appreciable over the scale of an atomic nucleus, so there is no question of them directly affecting any macroscopic mechanical processes. They are responsible, however, for various nuclear reactions in the course of which nuclear energy is, most commonly, transformed into electromagnetic energy (X- or gamma rays) and thermal energy.
All the other forms of energy one encounters at the microscopic, and even the macroscopic, level have their origin in electromagnetism. Some of them, like the electrostatic energy in a capacitor or the magnetic interaction between two permanent magnets, are straightforward enough scale-ups of their microscopic counterparts, and may allow for a potential energy description at the macroscopic level (and you will learn more about them next semester!). Many others, however, are more subtle and involve quantum mechanical effects (such as the exclusion principle) in a fundamental way.
Among the most important of these is chemical energy , which is an extremely important source of energy for all kinds of macroscopic processes: combustion (and explosions!), the production of electrical energy in batteries, and all the biochemical processes that power our own bodies. However, the conversion of chemical energy into macroscopic mechanical energy is almost always a dissipative process (that is, one in which some of the initial chemical energy ends up irreversibly converted into thermal energy), so it is generally impossible to describe them using a (macroscopic) potential energy function (except, possibly, for electrochemical processes, with which we will not be concerned here).
For instance, consider a chemical reaction in which some amount of chemical energy is converted into kinetic energy of the molecules forming the reaction products. Even when care is taken to “channel” the motion of the reaction products in a particular direction (for example, to push a cylinder in a combustion engine), a lot of the individual molecules will end up flying in the “wrong” direction, striking the sides of the container, etc. In other words, we end up with a lot of the chemical energy being converted into disorganized microscopic agitation —which is to say, thermal energy .
Electrostatic and quantum effects are also responsible for the elastic properties of materials, which can sometimes be described by macroscopic potential energy functions, at least to a first approximation. They are also responsible for the adhesive forces between surfaces that play an important role in friction, and various other kinds of what might be called “structural energies,” most of which play only a relatively small part in the energy balance where macroscopic objects are involved.
1 Plus a positron and a neutrino


8.1: Linear Momentum and Force
Learning Objectives
By the end of this section, you will be able to:
- Define linear momentum.
- Explain the relationship between momentum and force.
- State Newton’s second law of motion in terms of momentum.
- Calculate momentum given mass and velocity.
The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity.
Linear Momentum
Linear momentum is defined as the product of a system’s mass multiplied by its velocity:
\[p = mv \label{linearmomentum}\]
Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum \(p\) is a vector having the same direction as the velocity \(v\). The SI unit for momentum is \(kg \cdot m/s.\)
Example \(\PageIndex{1}\): Calculating Momentum: A Football Player and a Football
- Calculate the momentum of a 110-kg football player running at 8.00 m/s.
- Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.
Strategy
No information is given regarding direction, and so we can calculate only the magnitude of the momentum, \(p\) (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in Equation \ref{linearmomentum}, which becomes
\[p = mv \nonumber\]
when only magnitudes are considered.
Solution for (a)
To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation.
\[\begin{align*} p_{player} &= (110 \, kg)(8.00 \, m/s) \\[5pt] &= 880 \, kg \cdot m/s \end{align*}\]
Solution for (b)
To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation.
\[\begin{align*} p_{ball} &= (0.410 \, kg)(25.0 \, m/s) \\[5pt] &= 10.3 \, kg \cdot m/s \end{align*}\]
The ratio of the player’s momentum to that of the ball is
\[\dfrac{p_{player}}{p_{ball}} = \dfrac{880}{10.3} = 85.0 \nonumber\]
Discussion
Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.
Momentum and Newton’s Second Law
The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes.
Newton’s Second Law of Motion in Terms of Momentum
The net external force equals the change in momentum of a system divided by the time over which it changes.
\[F_{net} = \dfrac{\Delta p}{\Delta t}\]
where \(F_{net} \) is the net external force, \(\Delta p\) is the change in momentum, and \(\Delta t\) is the change in time.
Making Connections: Force and Momentum
Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics.
This statement of Newton’s second law of motion includes the more familiar \(F_{net} = ma\) as a special case. We can derive this form as follows. First, note that the change in momentum \(\Delta p\) is given by
\[\Delta p = \Delta (mv)\]
If the mass of the system is constant, then
\[\Delta (mv) = m\Delta v.\]
So that for constant mass, Newton’s second law of motion becomes
\[F_{net} = \dfrac{\Delta p}{\Delta t} = \dfrac{m \Delta v}{\Delta t}.\]
Because \(\frac{\Delta v}{\Delta t} = a, \) we get the familiar equation
\[F_{net} = ma\]
when the mass of the system is constant .
Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail ; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.
Example \(\PageIndex{2}\): Calculating Force: Venus Williams’ Racquet
During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?
Strategy
This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as
\[F_{net} = \dfrac{\Delta p}{\Delta t} \nonumber\]
As noted above, when mass is constant, the change in momentum is given by
\[\Delta p = m\Delta v = m(v_f - v_i). \nonumber\]
In this example, the velocity just after impact and the change in time are given; thus, once \(\Delta p\) s calculated, \(F_{net} = \frac{\Delta p}{\Delta t}\) can be used to find the force.
Solution
To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.
\[\begin{align*} \Delta p &= m(v_f - v_i) \\[5pt] &= (0.057 \, kg)(58 \, m/s - 0 \, m/s)\\[5pt] &= 3.306 \, kg \cdot m/s = 3.3 \, kg \cdot m/s \end{align*} \]
Now the magnitude of the net external force can determined by using \(F_{net} = \frac{\Delta p}{\Delta t}\)
\[\begin{align*} F_{net} &= \dfrac{\Delta p}{\Delta t} = \dfrac{3.306 \, kg}{5.0 \times 10^{-3}}\\[5pt] &= 661 \, N,\end{align*} \]
where we have retained only two significant figures in the final step.
Discussion
This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using \(F = ma\) but one additional step would be required compared with the strategy used in this example.
Summary
- Linear momentum ( momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
- In symbols, linear momentum \(p\) is defined to be \[p = mv \nonumber\] where \(m\) is the mass of the system and \(v\) is its velocity.
- The SI unit for momentum is \(kg \cdot m/s.\)
- Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
- In symbols, Newton’s second law of motion is defined to be \[F_{net} = \frac{\Delta p}{\Delta t} \nonumber \]where \(F_{net}\nonumber \) is the net external force, \(\Delta p\) s the change in momentum, and \(\Delta t\) is change in time.
Glossary
- linear momentum
- the product of mass and velocity
- second law of motion
- physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes


10.5: Fourier Transforms of Differential Equations
The Fourier transform is a useful tool for solving many differential equations. As an example, consider a damped harmonic oscillator subjected to an additional driving force \(f(t)\) . This force has an arbitrary time dependence, and is not necessarily harmonic. The equation of motion is \[\frac{d^2 x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega_0^2 x(t) = \frac{f(t)}{m}.\] To solve for \(x(t)\) , we first take the Fourier transform of both sides of the above equation. The result is \[- \omega^2 X(\omega) - 2 i\gamma \omega X(\omega) + \omega_0^2 X(\omega) = \frac{F(\omega)}{m},\] where \(X(\omega)\) and \(F(\omega)\) are the Fourier transforms of \(x(t)\) and \(f(t)\) respectively. To obtain the left-hand side of this equation, we used the properties of the Fourier transform described in Section 10.4, specifically linearity (1) and the Fourier transforms of derivatives (4). Note also that we are using the convention for time-domain functions introduced in Section 10.3.
The Fourier transform has turned our ordinary differential equation into an algebraic equation which can be easily solved: \[X(\omega) = \frac{F(\omega)/m}{- \omega^2 - 2 i\gamma \omega + \omega_0^2}\] Knowing \(X(\omega)\) , we can use the inverse Fourier transform to obtain \(x(t)\) : \[x(t) = \int_{-\infty}^\infty \frac{d\omega}{2\pi} \, \frac{e^{-i\omega t}\, F(\omega)/m}{- \omega^2 - 2 i\gamma \omega + \omega_0^2}, \;\; \mathrm{where}\;\; F(\omega) = \int_{-\infty}^\infty dt\; e^{i\omega t} f(t).\] To summarize, the solution procedure for the driven harmonic oscillator equation consists of (i) using the Fourier transform on \(f(t)\) to obtain \(F(\omega)\) , (ii) using the above equation to find \(X(\omega)\) algebraically, and (iii) performing an inverse Fourier transform to obtain \(x(t)\) . This is the basis for the Green’s function method, a method for systematically solving differential equations that will be discussed in the next chapter.


2.1: Why it Matters- Economic Environment
Why learn about fundamental economic principles?
Economics is about choices: namely, how we choose to allocate scarce resources and how our choices impact others. Beyond that, it’s about the choices made by businesses, government, and other countries. Let’s begin with ice cream—something familiar and tasty to most people—and use it as a framework for thinking about economics.
Let’s say you live in Boise, Idaho, and decide to go out for ice cream with your friend Charlie, who has been reading about ice cream flavored with Persian saffron. That’s the kind he wants, but none of the five ice-cream shops you visit has it. Why not? Why can’t they meet his demand? Because Charlie is the only person in Boise who has ever asked for it, and none of the shops has decided to carry something so expensive (the saffron costs nearly $1,000 per ounce) and, frankly, strange. They have chosen to offer ice cream that is low-cost, high-demand, and easy to manufacture.
The next day, Charlie is still obsessed with saffron ice cream, and he’s discovered a shop in Dubai, where, for the mere price of $816 (per scoop, fancy toppings included), one can try this exotic treat. Now Charlie has some economic choices to make. Including the $1,800 airfare, plus the other travel expenses (hotel, cabs, etc.), the trip to Dubai will cost $2,600—at least. It will also cost Charlie time: the time it takes to plan the trip, the days off work, and travel time. Then, there’s the opportunity cost. While he is planning, traveling, and eating his ice cream, Charlie’s giving up the opportunity to do other things with his time and money. And there are unforeseen choices and expenses: What if he gets to Dubai and they’ve run out of saffron ice cream? Does he hang around in Dubai, investing more resources and waiting for them to restock? What if he tries Persian saffron ice cream and says, “Yuck!”—and wishes he’d bought banana ice cream back in Boise?
This is an extreme, unlikely situation, of course. However, the choices that ice-cream manufacturers, ice-cream shops, and people like Charlie make every day are all examples of economic decisions: at every turn, a choice has to be made about the allocation of limited resources. What economic decisions do you make in your life?
The purpose of this module is to give you an understanding of the fundamental principles of economics, some of the factors that drive economies, and how economics shapes the business environment. You will likely learn more about economics as you continue your education, but this section should serve as an excellent introduction and give you some tools to think about the impact of economics on your daily life.


1.3: Arabic's Influence in the Region
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This section seeks to convey the importance of Arabic as a language which came to be spoken far beyond its place of origin, and which also gained influence on a large geographical area through the conquest of Islam. We also seek to address a unidimensional view of Arabic as the only linguistic influencer of the region. Indeed, Greek and Persian both had significant impacts on Arabic, both conceptually and in the form of many borrowed words. The major powers of the area when Islam came into being, the Eastern Roman, or Byzantine Empire, and the Sasanian Empire, spoke Greek and Persian respectively.
Arabic forms an initial basis for understanding many of the most important concepts shared by members of the larger Muslim umma, or Muslim community, within many diverse cultural communities. For example, the word sunna. A revered word when used in the context of the Prophet’s Sunna. Before the prophet sunna was a path worn into the sand, hardening the sand with perpetual use. It came to stand for tradition. In Bedouin culture, the nomadic culture of the Arabian Peninsula, such paths were very important, as unknown territory could be deadly. Over time, the word may have stopped emphasizing the original cultural reference, but it has kept the meaning of correct path. The term Sunni, as in Sunni Muslims, means those who are following the right path; in this case, the path refers to the traditions of the Prophet.
Arabic and many aspects of Arab culture spread worldwide through Islam; one could justifiably consider early Islamic conquests, Arab conquests. Before Islam, there was a vibrant literary culture in the Arabian Peninsula. The Ka’aba, now known for being the global center for pilgrimage, was already a pilgrimage site, and the place where the best poems were displayed or “hung”. One of the masters of these “Hanging Poems” was ‘Antara, and his poetry depicted a love relationship with ‘Abla. The picture on the previous page is supposed to show the rock they used as a meeting point in his poetry.
‘Antara’s significance lies in the significance of pre-Islamic Arab society. The language of the Qur’an refers to that cultural milieu a great deal, and many of the early Arab cultural practices inform Islamic concepts. Thus, remnants appear in Islamic law, religious and social practices, aesthetics, etc., in Muslim communities, regardless of whether they are native speakers of Arabic.
The cultural connections between Arabic and Islamic make it exceedingly easy to conflate “Arab” and “Muslim”. The largest Muslim-majority country in the world, and the fourth most populous country worldwide, Indonesia, is neither Arabic-speaking (as mother tongue), nor in the Middle East. As we pointed out in the Introduction, the Middle East is only a small portion of Muslims worldwide, and only 16 of the 25 countries in our geographical definition of the Middle East are Arab-majority countries. Thus, Arab-majority countries are all within the Middle East (according to our definition), but not all Middle Eastern countries – or Muslim-majority countries – are Arab (Israel is Jewish-majority).
There is some validity to the connection between Arab identity and Islam, however. The Qur’an is in Arabic; and the Sunna, or the authenticated accounts of the prophet’s words and actions, are in Arabic. Thus, the terminology in non-Arabic-speaking, Muslim-majority countries for them is in Arabic. Most Muslim-majority countries also have adopted the Arabic script, as well (Turkey is a notable exception). Standard Islamic expressions, such as the greeting, es-Salamu ‘Alaykum, or, Peace Be Upon You, remain in Arabic, regardless of cultural context.
Arabic was much less influential beyond the Arabian Peninsula and the Syrian Plateau prior to the advent of Islam. When Islam began, the Arabian Peninsula was flanked by two superpowers of Eastern Roman, or Byzantine Empire, and the Sassanid, or Sassanian Persian Empire. Greek was the predominant language of the former, Persian the latter. There is evidence that both Greek and Persian influenced early Arabic. Persian culture also influenced Greek culture (Miller, 2004). Thus, the exchange was multi-directional prior to Islam. Related map: “Indo-Sassanid” by Guillem d’Occam, via Wikimedia Commons: http://commons.wikimedia.org/wiki/Fi...o-Sassanid.jpg
Relics of the time before Arab dominance reside in the Arabic of the Qur’an and some remain in the language today. Persian-speaking and Greek-speaking super powers flanked the Arab communities (see map, right) that originally spread the language of Arabic, along with the religion of Islam. And those influences are apparent in many of the words, along with older semitic languages, such as Aramaic and Syriac. According to linguist and Arabic historian Kees Versteegh, there were borrowings from Persian, Greek, and Aramaic in the Qur’an and in pre-Islamic poetry. Some examples include (Versteegh,2001,p.60):
• the word for rose, warda , which came originally from Avestan Persian, vareda
• the word for money, or ‘silver coin’, dirham , originally from the Greek, drachme’
• the word for mosque, or masjid , originally from the Aramaic/Nabatean msgd’


2.34: Tips for the Listening Exam
In this exam, I would like you to ask yourself when you are listening to a piece: “What am I hearing?” Specifically listen for the musical characteristics that would enable you to recognize the genre (concerto, fugue, etc.) or individual movement (aria from an opera). One of the easiest ways to distinguish pieces on this listening exam is to identify whether the music is vocal or instrumental, so that’s where we’ll start. Remember the all important question: “What am I hearing?”
I Hear Singing
This immediately rules out several genres: Trio Sonata, Solo Concerto, Concerto Grosso, Fugue, and Suite. Those are instrumental genres. It must be an Opera, Cantata, or Oratorio. Unlike the instrumental pieces, knowing the genre won’t necessarily give you the piece. After all, an aria from an opera sounds the same as an aria from a cantata or oratorio. So in these paragraphs we’ll talk about how to pick out specific pieces by once again asking the question: “What am I hearing?”
I Hear One Singer with Continuo
There are two pieces that feature a singer with sparse accompaniment: “Tu se’ morta” and “Possente spirto” from Monteverdi’s L’Orfeo . The two vocal movements from L’Orfeo can be distinguished from each other by the fact that one movement is an arioso (“Tu se’ morta”) while the other is an aria (“Possente spirto”). You might be surprised an aria falls under the category “I hear one singer with continuo,” but in the early days of the Baroque, the concept of an orchestra was quite different from the large ensemble that we think of. Even the orchestras of Bach and Handel in the late Baroque, though small by modern standards, were much larger than those used in the earliest operas. The orchestra in L’Orfeo is composed of less than a dozen instruments and it punctuates the singer’s melody rather than accompanies it.
Speaking of the tiny orchestra, it provides an easy means of recognizing the aria: if you hear violins or trumpets interspersed with Orpheus singing you know you’re hearing “Possente spirto.” Also remember that because “Possente spirto” is an aria it’s going to show off the singer’s ability. That means you’re going to have ornamented melodies. The tune will be much more active and nimble than in “Tu se’ morta.” Also there is a particular vocal ornament that early Baroque audiences really liked but that we tend to think sounds a little silly – a kind of stuttering sound on a sustained pitch. However funny it may sound to our ears, it’s really hard to do (try it and see) so it was a sign to the audience that this was a very talented singer.  The arioso “Tu se’ morta” is tuneful but not nearly as ornamented as the aria. It also, because of the subject matter, is more somber and forlorn in its expression. It is only accompanied by continuo so you won’t hear any violins or trumpets playing as Orpheus makes his decision to descend into Hades in search of his beloved Euridice.
I Hear One Singer with an Orchestra
It must be an middle or late Baroque aria. You have two arias on this test that are accompanied by an orchestra: “When I am laid in earth” (Purcell’s Dido and Aeneas , an opera) and “Rejoice Greatly” (Handel’s Messiah , an oratorio). Arias’ main purpose is to express emotion, and the emotions of these to arias make them easy to distinguish. “Rejoice Greatly” expresses joy and happiness (at least during the first and last sections of the da capo aria) while “When I am laid in earth” expresses sorrow. Both arias are in English so the emotion being expressed in each piece will be easier to pick up on as you can rely on the words as well as the character of the music. That may be enough to tell these two apart, but in addition you can listen for the bass line. In “When I am laid in earth” there is a ground bass. As a result the bass line is a short, descending melodic idea that repeats over and over without change for the entire aria. “He shall feed His flock” does not have this feature, so the bass line will also help you tell these two similar pieces apart.
I mentioned earlier that “Rejoice Greatly” is a da capo aria. While it’s important to understand that musical characteristic (da capo arias are essentially in ABA form with the singer expected to ornament the melodic line in the second A section), it may not be evident in a one-minute excerpt. That said, I promise that I won’t throw you a trick question by playing only the B section from this particular aria since it’s the A section that expresses the emotion of the piece. You’ll hear either the A section (the part that actually says “rejoice greatly”) alone or portions of both sections.
I Hear a Choir with an Orchestra
There are three choruses on this test and they’re all from the same cantata by J. S. Bach. We have movements I, VI, and VII (that’s one, four, and seven) from Bach’s church cantata Wachet auf, ruft uns die Stimme . While at first it might seem like these would be hard to tell apart, these three movements are actually quite different from each other. One of the best things to listen for is how Bach treats the texture. While all three movements are polyphonic, Bach handles the polyphony quite differently each time. I’ll just go through them each in order.
Movement I features the most active and independent polyphony of the three. By that I mean that each voice part in the choir and each instrument in the orchestra has it’s own independent line that is woven together with the rest to create a glorious but very dense and complex musical fabric. For example, the sopranos sing a stately melody (the chorale tune) in long notes. Underneath them the altos, tenors, and basses race around with more active lines. All the while the violins, woodwinds, and low strings add their own lines to the elaborate mix. This is also the fastest of the three movements.
Movement IV has independent polyphonic lines too, but a lot fewer of them than movement I. There are only 3 lines woven together: the walking bass line played by the low strings, a beautiful countermelody played by the violins, and the main melody (the chorale tune) sung by the tenors. You’ll still be able to hear that this is a choir as there are a lot of tenors singing, but they all sing the same part. It’s basically a song for choir.
Movement VII is distinguished by a homorhythmic texture. That means that while there are four lines (polyphony) of music being worked together, they move more or less in lockstep with each other. The rhythm for each part (soprano, alto, tenor, and bass) is largely the same, hence the term homorhythm. This means that they all sing the words at pretty much the same time so even if you don’t speak German you can hear the words (unlike movement I where each part is singing different words at the same time). You’ll also notice that unlike the other two movements, the orchestra does not have independent lines. It plays the same notes as the singers. This is called “doubling.” Finally, movement VII is the slowest of the three.
I Hear Instruments Only—No Singing
If no voices are present in the piece you can immediately rule out Opera, Oratorio, and Cantata. Those are vocal genres. It must be one of the instrumental genres such as Trio Sonata, Concerto, or Suite. Because our Baroque pieces consist of one example from each genre, determining the genre will give you the correct answer. Often the kinds of instruments featured in the piece and how they are used will provide major clues as to the genre. As always, ask yourself “What am I hearing?”
I Hear Solo Organ
This has to be the Organ Fugue in G minor by Bach. This is another piece where the title also provides the genre: fugue. Now I suppose it is naive of me to think that anyone would listen for anything besides the solo organ instrumentation on the exam. Basically if you hear an organ by itself, it’s this piece. However, I do want to point out a major characteristic of a fugue so that if you were to come across one in the last 6 questions of a listening exam you’d be able to recognize it. A fugue is a complex style of polyphony based on the imitation of a single melodic idea: the fugue subject. This is what makes a fugue different from the imitative writing we heard in the Renaissance. The points of imitation we heard from Josquin and Palestrina were a series of different melodic ideas, each of which would be imitated once by every voice, or voice part (soprano, alto, tenor, etc.), in the choir. In a fugue there is only one central idea, the subject, that is imitated in all voices many times throughout the piece. That provides a real sense of unity and repetition in an otherwise highly complex and elaborate piece of music. That is in itself is something that you can recognize, but there is one more thing to be aware of. Because this subject is so important to the piece, a fugue always begins with the first voice presenting the subject all by itself. Notice how our organ fugue by Bach starts out with a single melodic line (the subject). Almost every fugue will do the same thing. So if you hear a melodic line presented all by itself (especially if it’s instrumental) which is then joined imitatively by successive voices or lines of music, you are probably hearing a fugue.
I Hear Solo Cello
If you hear a cello playing all by itself it can only be the Prelude from Bach’s Cello Suite No. 1 in G major. Once again, instrumentation will be the giveaway for this piece. Only one piece on this exam features the cello as the solo instrument so this should be a fairly easy piece to recognize. That said, I hope you will pay attention to the fact that this piece consists primarily of arpeggios—notes of a chord played in succession, one after the other, instead of all at once. Preludes were often written as a kind of warmup for more challenging pieces to follow. Often you’ll find composers pairing a prelude with a fugue, the prelude serving to help prepare the performer’s fingers and the audience’s ears for the more complex and challenging fugue to come. In this case the prelude paves the way for the subsequent dance movements of the suite. It is a testament to Bach’s talent that an introductory piece sounds so beautiful all by itself.
I Hear Solo Violin and Harpsichord
Correlli’s Sonata in B flat Major, Opus 5 No. 2: II Allegro is our only piece of chamber music on the exam. The solo violin should really stand out. Don’t get this confused with the solo violin concerto by Vivaldi. That piece features the contrast between a full orchestra and a solo violin. This is a sonata in which the only thing playing along with the violin is the continuo. And in this particular performance, the continuo is played by harpsichord alone rather than the usual cello/harpsichord pair. Remember that sonatas are a form of chamber music. Chamber music is intended for performance in a smaller space for smaller audiences. It is easy to imagine a group of aristocrats gathered in an elegantly appointed room in a palace listening to the performance of the violinist. A concerto features a much larger ensemble and would need to be performed in a larger space such as a concert hall.
I Hear Orchestra and Solo Violin
This has to be the 1st movement of Vivaldi’s “Spring” from The Four Seasons . This work is a solo concerto and as such features a contrast between a full orchestra and a soloist. They also are designed to show of the virtuosity of the soloist so you can expect to hear some really flashy playing from the soloist. Vivaldi doesn’t disappoint in that regard. He was a virtuoso violinist as well as a composer so he was writing music that would show off his abilities as a performer. If you hear orchestral music with virtuosic solo violin passages, it has to be Spring.
I Hear Orchestra with Concertino (Contrasting Small Group of Instruments)
This has to be Bach’s Brandenburg Concerto No. 5, 1st movement. This piece is a concerto grosso. Like a solo concerto it is based on the principle of contrasting groups of instruments. Instead of the orchestra vs. soloist contrast that you expect from a solo concerto, however, you have orchestra vs. concertino. The concertino could be made up of any instruments, generally 2 to 4, drawn from the larger orchestra. In the first movement of this piece you have a concertino of flute, violin, and harpsichord. Sometimes all the instruments of the concertino play together and at other times they alternate. This makes the concerto grosso a bit more flexible in its sound. One unusual characteristic of this particular piece is Bach’s emphasis on the harpsichord. Normally the harpsichord is in the background but Bach puts it center stage with some extremely virtuosic passages for the keyboard. In fact there are times when the harpsichord plays all by itself in this piece, almost as though it were a solo harpsichord concerto. Listen for the small group/large group contrast and the fiery harpsichord playing.
I Hear an Orchestra all by Itself, No Contrasting Soloist or Concertino
There is only one piece on your exam for full orchestra without any soloist or concertino: “Alla Hornpipe” from Handel’s Water Music . Remember what Handel was writing his Water Music for. It is a series of suites to be played in the presence of the King of England while on a royal barge trip (a parade really) on the River Thames. By the way, I know when I say barge you think of a big flat thing hauling coal or logs on the Mississippi, but these were really fancy boats. Anyway, this music was for an outdoor performance on a busy commercial waterway with only as many musicians as you could fit on the barge (not many). So it’s no surprise that Handel gave some really prominent passages to the loudest instruments: the horns, trumpets, and drums. If you hear a big orchestral sound with standout moments for the brass, you’re hearing Water Music . I should mention that in this movement you have an ABA form, with the prominent use of brass instruments found only in the A section. The B section is comparatively shorter, features active lines for the strings and woodwinds without brass or percussion, and is in a minor key to contrast the major key of the A section. As with other pieces that features a similar ternary form, I feel it would be misleading if your listening excerpt was taken solely from the B section. On the listening exam, if you hear a portion of this piece it will either come from the A section alone or will have both B and A sections represented.
Titles, Composers, & Genres for the Baroque Music Exam
Here is a list of all the titles, composers, and genres that you could encounter on the Baroque Music Listening Exam. They are listed just as they will be on the exam. Good luck!
| Title | Composer | Genre |
|---|---|---|
| Vocal Music | ||
| Tu se’ morta, L ‘Orfeo | Claudio Monteverdi | Opera |
| Possente spirto, L’Orfeo | Claudio Monteverdi | Opera |
| When I am laid in earth, Dido and Aeneas | Henry Purcell | Opera |
| Wachet auf, ruft uns die Stimme , I | Johann Sebastian Bach | Cantata |
| Wachet auf, ruft uns die Stimme , IV | Johann Sebastian Bach | Cantata |
| Wachet auf, ruft uns die Stimme , VII | Johann Sebastian Bach | Cantata |
| Rejoice Greatly, Messiah | George Frideric Handel | Oratorio |
| Instrumental Music | ||
| Sonata in B flat Major, Opus 5 No. 2: II Allegro | Arcangelo Corelli | Sonata |
| Spring, The Four Seasons | Antonio Vivaldi | Solo Concerto |
| Prelude, Cello Suite No. 1 in G major | Johann Sebastian Bach | Suite |
| Organ Fugue in G minor | Johann Sebastian Bach | Fugue |
| Brandenburg Concerto 5, 1st mvmt. | Johann Sebastian Bach | Concerto Grosso |
| Alla Hornpipe, Water Music | George Frideric Handel | Suite |
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution


1.37: Some Other Writers on Love
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Learning Objectives
- Recognize the problems with typical kinds of explanations about love which are generally less analytic, less logical, and/or less evidence-based.
Watch this video or scan the QR code to learn more about the philosophy of Aristotle.
I want to point out what I believe are some flaws in some particular points about love by other authors and also to point out what I think is an erroneous style of analysis of love. I believe some of these points and this style have been unduly influential. Others demonstrate how some views about love and about relationships can sound quite plausible at initial reading and yet still be seen to be erroneous under more reasonable scrutiny. I believe the proper approach to the subject is the analytic kind I have taken in this book, trying to put into a sensible and reasonable general perspective the kinds of feelings and experiences that are open to most people, and that happen to many; but a perspective that at the same time tries to reasonably take into account the realistic differences there are among people.
In their book Mirages of Marriage (W.W. Norton & Co., 1968), which is marvelous for its clear, concrete language and ideas, and for its many practical insights, W.J. Lederer and Don D. Jackson work with Harry Stack Sullivan’s definition of love (p. 42): “When the satisfaction or the security of another person becomes as significant to one as is one’s own satisfaction or security, then the state of love exists.” (From Harry Stack Sullivan’s Conceptions of Modern Psychiatry (W.W. Norton & Co., 1953, pp. 42-43.)
Though on the whole, I believe this is an excellent practical book with a number of theoretical insights that are worth considering, there are a few things I disagree with in it. One is this theoretical definition of love. I disagree for the following reasons:
(1) One might feel another’s happiness or satisfaction and security are extremely important — a counselor, teacher, doctor, or anyone might expend a great deal of time and energy and worry concerning a patient, student, or friend because he cares about other people. Indeed, one might hope all of us would care about each others’, or many others’, happiness very much. This state might be considered love, or loving, in a very general “Christian” or humanitarian sense, but it is not the kind of love in the general sort of romantic sense involving relationships that we are concerned with here. It has nothing to do with specific kinds of feelings of attraction people might have for each other, and it also fails, I think,
(2) in that it focuses only on the concern people have for a loved one’s well-being and not on how successfully or unsuccessfully one might actually be in bringing that well-being about. If you care about another’s well-being but are unable to provide any for them, then, as I have argued extensively, though they can be attracted to you and though they can care about you, they should not be said to love you, since their loving you involve your actually being good for them and making them happy, and happy in meaningful ways. Otherwise, all they could have (though that is considerable) is attraction, concern, infatuation, or charity (and in extreme cases, martyrdom). And, if all you get out of the relationship is concern about their well-being, you too, do not have love, but just that concern. You or they could even harm each other though you both were trying not to, and again there would be some sort of self-sacrificing care and concern in the relationship, but it would generally not be what we would want to characterize as a loving relationship.
(3) A person could be concerned about another’s well-being for selfish reasons because he is dependent on them for something important to him; this would meet the definition but would not be love.
(4) As I have argued extensively throughout the book, satisfaction and security are not the main or only goals of ethics or the good in life. A person might properly keep appointments though he would be more satisfied to sleep late in bed or carouse with friends. A person might want to heighten his sensitivity to the suffering of others in order to be a better person; couples may want to rear their children to be sensitive and caring about the needs of others and to be kind. But sensitivity often leads to sorrow and dissatisfaction; a case could be made that selfish, insensitive people tend to have the most satisfaction (and possibly even security, at least in terms of what money can buy), though the satisfaction is undeserved.
(5) It is not just happiness or satisfaction and security that we seek, but deserved happiness or satisfaction — satisfaction that comes in certain ways, such as through creativity, hard work, perseverance, thinking honorable acts, etc., or satisfaction that is the result of delight in something good and worthwhile, such as enjoying music, literature, etc. If we could be satisfied by taking a magic pill or pharmaceutical lotus leaf, that is not what we would want or consider to provide satisfaction in the right way.
(6) Some people do not care about whether their loved ones are concerned so much with their satisfaction and security; they simply appreciate the fact those loved ones help provide it, not out of concern, but just naturally, as an outgrowth of the way those loved ones are. If someone, say, likes wit and falls in love with someone whose nature includes wit, then it does not matter whether they are being witty “for” their loved one or just simply being witty because they like to be.  She likes to feel she or her character then somehow blends nicely with you or your character. And, in fact, this is the way I think love generally is, though I think she takes too far or expects too much from the idea of naturally blending and never wanting someone else to make some sort of sacrifice for her. (And in fact, she is very altruistic, however, and would make all kinds of sacrifices for you; she just does not want you to make any for her.) Perfect natural blends are not likely; even in the best relationships, there are probably some sacrifices by each partner.
(7) Not being a concerned and caring person about other people’s security and satisfaction says more about your general nature than about whether you are in love or not. I think everyone should consider other people’s satisfaction and security, though proper fairness to one’s self demands that you are not always self-sacrificing because of it.
(8) I think people can love from afar, loving someone who makes them more satisfied and secure and to whom they are attracted, without being able to, needing to, or even trying to reciprocate. Sometime such reciprocation might even be unwarranted. For example, I think a student could love a (married) teacher, but it might be inappropriate to act toward that teacher in the way one might want to act toward a peer one loved in the same way. One could love a television celebrity or an author who they find attracting and who makes a meaningful contribution to their life.
(9) Finally, (and this one is technical, but I think necessary to be dealt with in case anyone else tries to resurrect a kind of definition of love similar to Sullivan’s above.) If one is not particularly concerned about his own well-being, and is then equally unconcerned about the well-being of others, by the above definition he would love others. But terrorists who kill themselves with their victims could hardly be said to love their victims. For the definition to do even what they want it to, it requires not only for there to be at least equal concern for another as for one’s self, but that there should be some reasonably substantial, actual (positive) concern for both.
Aristotle
In Books (“chapters” actually) 8 and 9 of his Nichomachean Ethics , Aristotle (op. cit.) seem to have been concerned with many of the same questions I am. But my answers are for the most part different from his. For example, he thought affection had to have a motive, either (1) expected good or pleasure from the loved one or expected usefulness from the loved one for the gaining of good or pleasure. But since he thought usefulness changed with time and circumstances and that goodness pretty much did not, he thought that friendship based on usefulness was not likely to last and that friendship based on gaining goodness would. Aristotle also thought that “like attracted like” with regard to good people and so their actions and being alike (and relatively stable so) and their seeking each others’ welfare because they were good, would cause their friendship to last a long time. However,
(1) I do not believe affection needs a motive. I think we become enamored of people often for unknown reasons or causes if any; perhaps sometimes more because of our own states of mind at the time than because of anything they really offer. Consider, for example, the phenomena of a person’s being attracted to someone who is very much like a person who he or she was not attracted to just a bit earlier (and) even though he or she knew the first person might have done him more good, given more pleasure, or have been of more use.
(2) Often someone can actually be bad for us or of no use in promoting our well-being, and we can know this and yet still be attracted to him or her. (Rollo May, for example, on page 283 of Love and Will (May, 1969) seems to think instant attraction is because (?) “the suddenly beloved elicits a composite image from our experiences in our past or in our dreams of our future; we spontaneously experience him or her in relation to our personal ‘style of life’ which we form and carry with us all our lives and which becomes clearer the more we know ourselves.” I think I’ve become fond or enamored of and attracted to people without all that happening, though I am not really sure what experience is characterized in the second part of that statement.)
(3) Usefulness is not necessarily so transient a thing. Many husbands and wives probably operate together a long time simply because they find it more efficient and convenient to remain a “team”. Many workers and their managers can be useful to each other in their entire careers. But, also
(4) it seems many people can be useful to us and we yet do not necessarily have an affection for them or their use; for example, an unfriendly boss-employee relationship or an addict’s feelings toward a pusher and habit he despises.
(5) I see no reason bad people cannot like each other or be lasting friends.
(6) It is not clear there are any (perfectly) good people in Aristotle’s sense or that if there are more than one, they act alike anyway or that they would necessarily be attracted to each other, or that like attracts like in general. (Research seems to show that among people, in general, it is often that opposites attract as well as “likes”; often it is people with complementary personalities or qualities that attract each other also.)
These last two points, however, would require extensive analysis of Aristotle’s notion of the good man, and that is not feasible here.
Others I have seen speak to the kinds of issues I have been concerned with in this book are various clergymen speaking at weddings or about marriage, Abigail Van Buren, Ann Landers, some Readers’ Digest articles, the “Playboy Advisor”, Dr. Ruth Westheimer on television, etc., though I have sought in this book to give a far more reasoned, clear, sustained, and supported account than any of these.
But I have found these kinds of things more helpful, when correct, than, for example, many kinds of professional philosophic or psychoanalytic works on the topic of love. One philosophical difference of opinion (between Robert Solomon and Hugh T. Wilder in Alan Soble’s Philosophy of Sex ) involves whether, if sex is a form of communication, masturbation then is like writing a letter but not mailing it, like talking to oneself, or like thinking aloud or writing for one’s own pleasure or clarification of ideas. I think this kind of argument is somewhat silly and misses the real nuances and points of masturbation. Further, they seem to miss the more important point that sex, as I have extensively argued, is not a meaningful form of communication — even when it is interpersonal.
Though Erich Fromm, in his Art of Loving (Fromm, 1956), does have a section on the practice of love, the book is far less practical than might have been hoped. Fromm’s guidelines of discipline, concentration, patience, concern, overcoming of narcissism, faith, reason, etc. hardly tell one much about whether a particular relationship he or she is involved in is good or not, partially good or not, or what. I think these points, though useful, need to be at least elaborated.
I also think more needs to be said about what Fromm calls pseudo-loves, why they are bad or wrong or not really love at all. In fact, I think it is a mistake simply to label them pseudo-love, and better to say they are relationships which some call love, which others do not, but which are bad or good because…or that could be impoverished or improved by.
Fromm states “every theory of love must begin with a theory of man, of human existence.” He then goes on to describe man’s anxiety arising from his separation from the rest of the universe and others in it, and love as a solution to that anxiety problem. I do not exactly know what a theory of man or human existence is; but I do think that to talk about human relationships, whether meaningful, loving, friendship, or of any sort, one must know some things about people. If this knowledge constitutes a theory about man (which I doubt), fine; but I do not think one needs some (other sorts of) theory of man to talk sensibly about love any more than one needs a theory of man to be a good football coach (and I do not mean just a winning football coach), teacher, welfare worker, nurse, policeman, or whatever. One does have to be sensitive, have some understanding of people, be discerning, etc., but not necessarily have some (one) theory about the human condition.
Further, to say that love is an answer to a problem — here the problem of man’s anxiety due to separation — is to give a use for love, not a description of it. Love is not explained or described by saying it is a cure for anxiety any more than water or Gatorade is explained or described by saying it is a cure for thirst.
And I am not certain that love is the answer to anxiety anyway. First of all, surely there can be love without anxiety; there seem to be people who love and are loved who were not particularly anxiety-ridden. But moreover, there are people who are loved who are still tremendously anxiety-ridden. In fact, many people are anxiety-ridden because they worry very much about their families and loved ones. Of course, the world would be nicer and less anxiety-producing if more people or everyone were loving, but that is just as true if more people or everyone were understanding and ethical, including being properly compassionate. Again, that may be a way of describing a kind of Christian or humanitarian love, not the kind specific to more personal or intimate relationships. But even that kind of humanitarian love would not particularly eliminate or “cure” anxiety due to fear of physical or non-moral catastrophe — terminal illness, severe birth defect, accident, earthquake, tornado, etc., though it would go a long way in some cases in making such catastrophe perhaps a little more bearable. (In the Birmingham, Alabama Museum of Art, there once was an exhibit of art, particularly paintings, whose content had to do with medicine. Some of the 19th-century paintings depicting the helplessness of grieving and compassionate physicians besides dead or dying patients almost seem to make that a more tolerable situation than those of the twentieth century depicting the more often successful technological, but impersonal, the awesomeness of contemporary medical resources. Certainly, both advanced medical resources and compassion are important, and in some instances, the two can be found together. But not often enough. And at least in all those cases where death is unavoidable or unpreventable anyway, the compassion and understanding are far more important than the technology. Still, this compassion and understanding are not love, even when small parts of it.)
Fromm believed that capitalism is a deep part of the lack of love and disintegration of it, and it may be to the extent that a capitalistic society can let money and profit seem more important than anything else, including relationships. But relationships occur whenever people are in contact with each other and some relationships are better or worse than others, and some have feelings and aspects other relationships do not have. Some of these will then be what I consider love. And it seems to me all this would occur no matter what the economic system. It is important to talk about what kinds of aspects and feelings are good and ought to be cultivated. Economic, political, and social systems and conditions certainly influence people and their relationships. So do many other things, such as health, job satisfaction, etc., but until we have an understanding of what relationships really involve, it is difficult to understand exactly how these systems and conditions influence them or what kinds of things need to be done, perhaps within the system, to counteract the corrupting and disrupting forces. For example, it seems to me that lack of understanding of ethics and moral reasoning skills (as I explained them in the chapter on ethics) accounts for many of the problems people have in relationships, but I do not see how capitalism, as such (or by itself), can be blamed for that lack of knowledge. The attainment of maximum financial profit may be incompatible with the attainment of other values some times, but people can often opt, even in a capitalistic economic system, for achieving those other values at the expense of greater profit. Many people do that who, for example, pass up job promotions requiring relocation in order to stay in a city they and their family really like. Capitalism, whatever its failings, does not, by itself, force anyone to choose money over more important values. Insensitivity to, or ignorance of, that fact or those more important values, combined with capitalism may cause someone to choose profit over more important values and force that choice on dependent employees; but ignorance and insensitivity, it seems to me, can occur in any system and cause or allow people to choose or force on others values that are worse than possible alternatives. And I would think that kind of ignorance and insensitivity could be battled in a capitalist society as well as in any other. I doubt that any economic system just by itself is the cause or remedy of many social problems or of ignorance in important areas.
In the chapters on ethics, I explained why I thought psychoanalytic theories were in general suspect since other kinds of therapies often work better and since the kinds of stories that psychiatrists invent to lend credence to their theories (often) seem to be simply untestable, inventive fabrications. Freud’s specific idea that love is “aim-inhibited sex” or the result of a blocked sex drive seems false, given the ample historical, sociological, and anthropological evidence of persons and societies where there is love in spite of sexual liberalism and fulfillment. (See J. Richard Udry (1966), The Social Context of M arriage , Lippincott, specifically pp. 184ff.)
Rollo May
Although Rollo May’s book Love and Will (op. cit.) have some good points in it, I am disappointed by many of the kinds of arguments it contains.
(1) Dr. May argues from word etymologies, which at best show what ancient civilizations and word “coiners” believed, not what is necessarily true or even reasonable.
(2) He argues from (often psychoanalytic) interpretations of Greek myths and biblical stories which at best give interesting interpretations to such stories and perhaps show what early people believed but do not, therefore, give evidence of the truths which May claims. Some interpretations he gives seem a bit far-fetched anyway: concerning Jacob’s wrestling with the angel and his incurred thigh injury May says, “…he limped away from the scene; he is now a cripple.  He says of Eros ‘ bringing about fertilization of the earth by shooting an arrow into the barren ground that the arrow is phallic. But how else would a Greek myth show vegetation coming into being — not bullets shot into the ground, not some hero or god doing the menial work of planting? That pretty much leaves something like arrows or spears. One might ask then whether arrows rather than spears show feelings of sexual inadequacy…or perhaps confidence!
Often, anyway, his passages do not even show what he says they do. There is no way the following passage can fit his comment about it concerning the three aspects of time — past, present, and future.
“Once when ‘Care’ was crossing a river, she saw some clay; she thoughtfully took up a piece and began to shape it. While she was meditating on what she had made, Jupiter came by. ‘Care’ asked him to give it spirit, and this he gladly granted. But when she wanted her name to be bestowed upon it, he forbade this and demanded that it be given his name instead. While ‘Care’ and Jupiter were disputing, Earth arose and desired that her own name be conferred on the creature, since she had furnished it with part of her body. They asked Saturn to be their arbiter, and he made the following decision, which seemed a just one: ‘Since you, Jupiter has given its spirit, you shall receive that spirit at its death; and since you, Earth, has given its body, you shall receive its body. But since “Care” first shaped this creature, she shall possess it as long as it lives. And because there is now a dispute among you as to its name, let it be called “homo”, for it is made out of humus (earth).'” (pp. 290-291)
Of this, May says (p. 291):
“This … shows the realization of the three aspects of time: past, future, and present. Earth gets man in the past, Zeus in the future; but since ‘Care’ first shaped this creature, she shall possess it as long as it lives,’ i.e., in the present.”
But Earth does not get man in the past with Zeus getting him in the future. The passage says Earth gave its body and will get it back, just as Jupiter gave its spirit and will get it back. The time sequences are exactly parallel; both had something in the past and will regain it in the future. Further, man’s life is not some one-dimensional point in time, but itself has a past and future, so “Care” has man in more than just some “present”. This kind of error would be of minor importance except that May then goes on to try to use it to make important points.
“This excursion into ontology makes it clearer why care and will are so closely related, indeed are two aspects of the same experience…”
In a similar example, he says (p. 79), quoting Socrates from the “Symposium”:
“Those who are pregnant in the body only, betake themselves to women and beget children—this is the character of their love; their offspring, as they hope, will preserve their memory and give them the blessedness and immortality which they desire in the future. But souls which are pregnant — for there certainly are men who are more creative in their souls than in their bodies — conceive that which is proper for the soul to conceive or contain. And what are these conceptions? — wisdom and virtue in general. And such creators are a poet and all artists who are deserving of the named inventor.”
But he says of this:
“The Greeks also knew that there always is a tendency for eros to be reduced simply to sexual desire— epithymia or lust in their terms. But they insisted that the biological is not denied but incorporated and transcended in eros .”
Yet Socrates is simply stating that man seeks to become immortal in two ways, through biological and/or through intellectual creativity. There is no mention of one’s being better than the other or of one’s transcending or incorporating the other.
(3) Dr. May holds that the problems of the neurotic and the vision of the artist are prophetic for society but he offers no way to distinguish truly prophetic visions or insights from ones which do not come true. He offers no suggestion for distinguishing between neurotic problems which (a) will become symptomatic of society at large, as the environment, which is supposedly disturbing sensitive neurotic people, now become stronger and supposedly more overwhelming to everyone, and ones (b) which are simply individual problems or problems only of neurotics and which will never become problems faced by larger numbers of people. He only gives examples of past visions or neurotic difficulties that have shown themselves as problems of much of society later. Are we to believe there were no “false” artistic vision and no neurotic problems which are not prophecies of normal states to come!
(4) He argues from literature; but, again, just citing other people who (seem to) hold your views does not show your views to be true or even probable.
(5) He cites analogies from such things as Mozart’s music and from the animal kingdom which are interesting but which, as evidence for his views, are often simply amusing.  Apart from recent research reports that this is not true about the praying mantis, the fact that sex and death in this fashion is the most minute exception to the world’s creatures does not seem to matter. That the male bee and (perhaps) praying mantis die during or shortly after intercourse does not seem particularly relevant even to antelope, let alone to humans.
He argues here that it is death and the knowledge of death that makes love possible, that heightens love. I disagree. Certainly, the awareness of death helps us to take things which are dear less for granted. But these things must already be things which are dear to us, otherwise, we would not care whether we lost them or not.  The beauty of love is prior to the acute awareness of its potential loss. I would turn a phrase and probably be in agreement with Erich Fromm here that it is not death or the thought of death which makes love possible, but it is love and friendship which make death and the thought of death sometimes acceptable or at least not so lonely and terrifying an experience. The fact that love and death go together in so many stories that May points to is perhaps because death is a permanent type of parting (afterlife notwithstanding), and parting from a loved one evokes very poignant emotion, something that literature often tries to do.
(6) May argues such things as:
“The fact that love is personal is shown in the love act itself. Man is the only creature who makes love face to face, who copulates looking at his partner. … This opens the whole front of the person…all the parts which are most tender and vulnerable—to the kindness or cruelty of the partner. The man can thus see in the eyes of the woman the nuances of delight or awe, the tremulousness or the angst; it is the posture of the ultimate baring of one’s self.” (p. 311)
Let me parody that to:
The fact that having one’s teeth examined, cleaned, drilled, and fixed is a personal experience is shown in the act itself.  This opens the whole front of the person, all the parts which are the most tender and vulnerable, to the kindness and cruelty of the dentist and patient. The dentist can thus see in the eyes of the patient the nuances of delight or awe, the tremulousness, or the angst; it is the posture….
Let me also add, as a friend pointed out to me, that many times, particularly in the past, it was safer and far less trusting and vulnerable to be turned toward another than to have your back turned to him.
And further, anyone with any kind of sensitivity can usually tell, from almost any angle or direction, and often from some distance away, when someone is afraid or anxious. One does not always have to look into another’s eyes to tell. On a date, one can often tell by the feel of how someone holds your hand whether they are doing it perfunctorily or passionately.  There are ways besides looking into someone’s eyes (if people even do look into each others’ eyes during intercourse) to understand how they feel.
(7) He spends pages arguing that sex without passion is an empty experience and that man does not see this and is heading toward the practice of apathetic sex. But his arguments are of the above sort, and my students in undergraduate introductory philosophy made the point independently and far more succinctly, forcibly, and reasonably in class that sex without love can be, and generally is empty. They seemed to see that. Further, in contrast to May (I think), it is not always the case that sex without long term passion or without love is empty; there are probably plenty of people who can generate sufficient short-term horizontal passion to make it a good experience for them even though they have little or no “general” passion toward the partner. Further, sex with a stranger or new acquaintance can be very exciting and reassuring. And sex with a friend or new acquaintance can be very comforting at a particular time or very important in any of a number of ways without being sex between two long-time passionate lovers. And there are plenty of people who can essentially (mutually) gratify each other for the sheer physical pleasure they get or cause—especially if they both know and accept their own and each others’ motives. Sex may have many motives besides love, and not all those other motives prevent it from being a wondrous experience. Nor, of course, does love insure that it will be a wondrous experience.
(8) May thinks that the Master-Johnson researches “are symptomatic of a culture in which the personal meaning of love has been progressively lost,” when in fact the research has as one goal helping people lead better and more loving lives through having fewer problems of sex and fertility. There is no claim by Masters and Johnson that all of the love can be reduced to sex, nor even that all of the sex (meaningfulness, emotional closeness, fantasy, psychology, etc.) can be reduced to sexual physiology.
(9) Some passages do not even seem to have a clear meaning, though they sound very nice or poetic. “We are in eros not only when we experience our biological, lustful energies but also when we are able to open ourselves and participate via imagination and emotional and spiritual sensitivity, in forms and meanings beyond ourselves in the interpersonal world and the world of nature around us.
“ Eros is the binding element par excellence. It is the bridge between being and becoming, and it binds facts and values together. Eros , in short, is the original creative force of Hesiod now transmuted into a power which is both ‘inside’ and ‘outside’ the person. We see that eros has much in common with the concept of intentionality proposed in this book; both presuppose that man pushes toward uniting himself with the object not only of his love but his knowledge. And this very process implies that a man already participates to some extent in the knowledge he seeks and the person he loves.” (p. 79)
All in all Love and Will is the kind of writing that I think helps prevent clear understanding and dialogue about love and relationships, rather than the kind that helps promote them. It is the sort of writing that, because it refers to many kinds of things that today only scholars tend to know about, sounds very scholarly, and then may intimidate people who cannot address the topic in those kinds of terms and who do not realize you do not have to.
Key Takeaways
- If you compare the analysis in this text with that of other authors, you will see most leave out crucial considerations.
- Some views about love and about relationships can sound quite plausible at initial reading and yet still be seen to be erroneous under more reasonable scrutiny.
Key Terms
- Eros (as defined by Rollo May) “is the binding element par excellence. It is the bridge between being and becoming, and it binds facts and values together. Eros , in short, is the original creative force of Hesiod now transmuted into a power which is both ‘inside’ and ‘outside’ the person.”
Review Questions
- Question : What were the motives Aristotle thought affection had to have?
- Question : What problems can there be about using myths to explain the nature and significance of sex?
- Question : What are some problems with Harry Stack Sullivan’s definition of love: “When the satisfaction or the security of another person becomes as significant to one as is one’s own satisfaction or security, then the state of love exists.”


15.4: Physiologic Adaptations during Labor and Birth
By the end of this section, you will be able to:
- Explain the physiologic changes of the person during the first stage of labor
- Explain the physiologic changes of the person during the second stage of labor
- Explain the physiologic changes of the person during the third stage of labor
- Explain the physiologic changes of the person during the hour after birth
The pregnant body undergoes a great deal of change during labor and birth, including many physiologic changes that enable the fetus to maneuver through the birth canal and be born. These physiologic alterations differ in each stage of labor and should be monitored by the nurse while the person is laboring and birthing.
Physiologic Adaptations during the First Stage of Labor
The shift from term pregnancy to the first stage of labor occurs when contractions lead to cervical change. During this shift, the body also prepares for the changes necessary for labor, birth, and the postpartum period. The alterations in nearly all body systems allow for the focus to move from pregnancy to birth during these hours or days. The nurse needs to understand how labor and birth impact body systems to be able to anticipate normal changes and identify deviations from normal that indicate the need for nursing intervention or consultation with the provider.
Vital Signs
The laboring person’s heart rate increases during contractions, and the baseline can increase or decrease. Sustained increases in heart rate warrant an investigation for possible signs of infection and excessive blood loss.
Blood pressure (BP) increases during contractions and returns to baseline between contractions. Slow increases in BP can be noted with the experience of pain; however, significant increases (>140/90 mm Hg) should be investigated for the presence of hypertensive disorders of pregnancy, which can occur during pregnancy, labor, birth, and the postpartum period (ACOG, 2020). Any hypotension should be investigated as possible infection, anesthesia side effects, or a sign of a concealed hemorrhage.
The laboring person’s temperature can increase with exposure to misoprostol (Cytotec) as well as epidural anesthesia , but any increase above 100.4° F (38° C) warrants investigation for possible infection.
The respiratory rate can increase, especially in unmedicated labor due to breathing techniques used as a coping mechanism.
The laboring person’s oxygen saturation can decrease with epidural anesthesia, especially if the medication gets above recommended levels, or when magnesium sulfate is administered for preeclampsia .
Fetal Descent
The pregnant person’s body continues to facilitate the labor process with contractions causing downward pressure on the presenting fetal part, which leads to descent through the pelvis. This typically occurs with internal rotation of the fetus combined with the physical pressure of uterine contractions and changes in pelvic diameters through maternal positioning in labor. During the first stage, the fetus is typically not engaged in the pelvis with a station of −1 to −3, but can be lower if fetal head engagement occurred prior to labor or rapidly during labor (see Figure 15.3(c)).
Cardiovascular
Cardiac output increases an additional 10 percent to 15 percent in the first stage of labor (Martin et al., 2022). This is likely related to increased stroke volume from the sensation of pain and/or the work of the body even when adequate anesthesia is present. The body is also protecting vasodilation around the uterine muscle to allow for perfusion of the fetus during the work of labor, which will need to abruptly switch to vasoconstriction after birth.
Musculoskeletal
Pelvic floor anatomy is stretching to accommodate the descent of the fetus in labor. The pelvis is affected by relaxing and position changes to increase diameters and allow passage of the fetus (Cohen & Friedman, 2023). The sacrum will often become more pronounced, especially when the laboring person is in an upright position, as the presenting part of the fetus engages into the pelvis and places internal pressure on the sacrum.
Gastrointestinal
Gastric emptying and motility are further slowed during labor. Slowed gastrointestinal function combined with intense pain and abdominal pressure can increase the likelihood of nausea and vomiting in labor, especially in the active phase of the first stage of labor. This can explain why many people in labor without anesthesia do not have the desire to eat, but those with epidural anesthesia may experience hunger. The physical exertion of labor requires nutritional support, and current recommendations are to consider oral hydration, electrolyte support through oral or intravenous replacement, and light solid food intake. The oral intake of hydration and nutrition is controversial in labor due to a history of fear surrounding the risk of aspiration in the rare event that general anesthesia is indicated during labor. Restriction of oral intake is not currently supported by research for low-risk labors, but protocol has been slow to change in many anesthesia orders and facilities (Singata et al., 2013).
Physiologic Adaptations during the Second Stage of Labor
The second stage of labor requires extensive maternal effort and fetal tolerance to achieve a spontaneous vaginal birth. To support this, the body prepares to adapt to these needs with certain changes. Contractions may space, or pause, to allow for greater rest between pushing efforts, blood pressure and heart rate increase for increased perfusion potential, the pelvis must remain mobile with consistent changes to allow for passage of the passenger , and other body processes may slow to divert energy to necessary functions for birth. Nurses can support patients by promoting these changes and offering reassurance of normality as well as monitoring for any pathologic changes that are outside the physiologic ones.
Contraction Pattern
Contractions during the second stage can space, or pause, immediately prior to the urge to push or during pushing efforts. This may be protective of the fetal acid-base balance. The increased force of the contractions combined with maternal expulsive efforts decreases oxygen perfusion to the fetus.
Vital Signs
In the second stage, blood pressure increases by another 10 mm Hg during active pushing, and both heart rate and respiratory rate increase as well.
Cardiovascular
As noted precedingly, heart rate and blood pressure increase in the second stage. Maternal exhaustion can be impacted by this, and appropriate rest or breaks between pushing efforts may be necessary if exhaustion or abnormal vital signs occur.
Musculoskeletal
Continued position changes and shifts in pelvic diameters or change in fetal position allow for fetal descent following the cardinal movements of vaginal birth. The birthing person will also often have an adrenaline surge, causing significant shaking or tremors during the end of the first stage and second stage of labor (Gicheru et al., 2019). These appear similar to chills experienced during a fever and will typically resolve in 1 to 2 hours after birth.
Gastrointestinal
Continued slowing of gastric emptying could be a contributing factor to the nausea and vomiting experienced during the second stage of labor. Increased acidity of the slowed gastric contents may lead to acid reflux or “heartburn” being reported by the birthing person. Epidural use is associated with less slowing of gastric emptying and may contribute to lower risk for aspiration in the use of general anesthesia (Bataille et al., 2014).
Physiologic Adaptations during the Third Stage of Labor
After birth of the fetus has occurred, the body makes significant shifts to accommodate for and limit blood loss. The body suddenly shifts from perfusing the fetus through the time of highest need to rapid vasoconstriction of the pelvic vasculature to prevent hemorrhage (Smith, 2020). The presence of the placenta prevents the completion of this process. Until the placenta is expelled, it interrupts vasoconstriction and prevents necessary hormonal shifts.
Uterus and Cervix
The term uterus is significantly distended and well perfused with purposeful vasodilation through the second stage of labor. The delivery of the fetus and drainage of remaining amniotic fluid that occurs with birth results in a rapid decrease in the internal pressure exerted on the uterine wall from within. In response to this, there is rapid vasoconstriction of the vessels that supply the pelvic floor. The 500 mL of blood flow routed to that area during labor is shunted back to the central circulation to compensate for expected blood loss of the same amount (Smith, 2020). Uterine contractions continue to shorten the muscle fibers and further decrease the uterine size, which can lead to placental detachment and subsequent delivery. The cervix will decrease in dilation and effacement as well but will remain partially dilated until the placenta delivers, typically around 5 cm dilation/50 percent effacement at this stage (Martin et al., 2022).
Cardiovascular
Rapid decrease in heart rate and blood pressure to prelabor levels may be noted or can occur slowly over the first 2 weeks postpartum. Hypotension and tachycardia are late symptoms of significant blood loss and warrant precise identification and treatment of the site of the bleeding. The shift of blood volume following uterine involution can increase the risk of cardiac complications such as cardiomyopathy and pulmonary edema, especially in those with preexisting heart conditions or hypertensive disorders in pregnancy (Martin et al., 2022).
Medications Prescribed during the Process of Labor and Birth
Medications prescribed for the discomfort of labor and birth are classified as analgesics and anesthetics. Both of these classifications are described in detail in 17.2 Pharmacological Pain Management.
Medications prescribed for the induction or augmentation of labor and to manage postpartum bleeding are classified as uterotonics and are described in detail in 18.3 Nursing Care During the Third Stage of Labor.
Physiologic Adaptations during the Fourth Stage of Labor
The fourth stage of labor is a very vulnerable time for the birthing person due to the rapid shifts occurring in many body systems as well as the risk for postpartum hemorrhage. Postpartum hemorrhage is one of the leading causes of maternal morbidity and mortality across the world. Postpartum hemorrhage is defined as blood loss greater than 1,000 mL following childbirth (ACOG, 2017b). See 20.1 Physiologic Changes During the Postpartum Period and 21.2 Postpartum Hemorrhage for more in-depth nursing care and management of postpartum hemorrhage. The nursing role of monitoring and educating during this vulnerable time is one of the most crucial tools available in reducing the risk of adverse outcomes in birthing people.
Vital Signs
In the fourth stage, the blood pressure and pulse may be slightly elevated or return to normal Significant or symptomatic decreases in blood pressure and heart rate call for careful assessment for blood loss, which can be overt or concealed. The respiratory rate returns to normal. Temperature may be slightly elevated (up to 100.4° F or 38° C) or normal (Martin et al., 2022).
Uterus and Cervix
Uterine involution continues during the fourth stage of labor. The fundus is expected to be firm, midline, and located near the umbilicus. Cervical dilation resolves, leading to a closed cervix shortly after delivery of the placenta (Martin et al., 2022).
Lochia
Bright or dark red lochia are expected for the entire fourth stage of labor. Continued measurement of this bleeding for a total quantitative blood loss can be useful in monitoring for the potential need for intervention. Quantitative blood loss is measured via calculation of the weight of blood-filled material with the weight of the material subtracted. Fundal assessment should produce only a small amount of bleeding. If continued leaking of streams of blood is noted or large clots are expressed during the fundal assessment, prompt consultation, increased monitoring, and interventions in collaboration with the health-care provider are indicated (Martin et al., 2022).
Perineum
Tenderness, edema, and a burning sensation along lacerations are to be expected. Topical sprays, ice, rest, anti-inflammatory medications, and use of a peri-bottle during urination to dilute urine can be useful tools to promote comfort during this time. Significant bruising or severe pain should be evaluated by the provider because of the risk for hematoma formation. Perineal sitz baths can also be utilized for pain control and promotion of healing. These are also available commercially if patients prefer to use them at home after discharge.
Bladder
Some postpartum patients experience decreased bladder sensation after birth. In those cases, the nurse should encourage the patient to attempt to void at regular intervals until the urge to void returns. The nurse should evaluate the patient’s bladder for distention, especially after epidural use. Some increase in voids can be seen as the body eliminates excess fluids in the first 2 weeks postpartum. Any bladder distention or decrease in urine output should be promptly reported to and evaluated by the provider. A distended bladder is one of the leading causes of postpartum uterine atony and can lead to postpartum hemorrhage (Martin et al., 2022).
Musculoskeletal
The work of labor can cause significant aches and pains that should be treated with rest, ice or heat, and support for ambulation as needed (Martin et al., 2022). If epidural anesthesia was used, it may be several hours before the birthing person is able to ambulate without assistance. The nurse should be sure to assess fall risk and call for assistance when supporting ambulation the first time after an epidural is discontinued.
Gastrointestinal
Bowel health can generate intense fear in the birthing person. The nurse should support bowel health with oral hydration, quality dietary intake with appropriate fiber, and the administration of a stool softener for any person with severe perineal tearing, hemorrhoids, cesarean birth, or those using narcotic pain medications (Martin et al., 2022).
Differentiating between Normal Adaptation and Early Warning Signs of Complications
Take action: Identify transition to tachycardia and assess for other chorioamnionitis symptoms.
When caring for a patient in the second stage of labor, an increase in heart rate can be expected due to the significant maternal effort required. When this increase becomes sustained and above 120 bpm, the nurse notes the change in the maternal vital signs. The nurse then obtains a full set of vitals, including a repeat temperature, and assesses the fetal heart rate for any changes as well. The maternal temperature is now 100.6° F with a heart rate of 140. The nurse discontinues pushing efforts and calls the provider to report the change in vitals. The provider orders an IV fluid bolus, a complete blood count (CBC), and antibiotics. The provider also orders the nurse to resume pushing efforts while the provider is en route to bedside management of the patient because this second stage of labor has increased in complexity.


1.5: The Start of Art - Overview
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How did humans initiate their desire to create art, and why? Undoubtedly, this query will never resolve itself by cultural anthropologists; however, the more archeological sites discovered and excavated, the more our discovery of humans, and their creativity continues. When an ancient settlement was established, archeologists take great care in removing artifacts, piecing together what the art meant to the culture. When foundational ruins from a magnificent building are discovered by archeologists, researchers study architecture, imagining how the significant buildings were designed and built without the aid of today’s technology or heavy equipment. How were the great pyramids constructed? How were the stones moved to build Stonehenge?
The definition of the word "prehistoric" generally means before the written word by humans. Does that make the history any less valuable? No. The definition, ‘pre’ means before the first evidence of written language. Maybe there is evidence not yet found about written language. How would that change the definition? Prehistoric times are divided into two periods: Paleolithic beginning about 2.6 million years ago to 12,000 BCE; and the Neolithic beginning 12,000 BCE and continuing until to 4500 BCE. The Paleolithic period experienced climate and geographic changes that ultimately affected all human societies on earth. Based on the tools found in the archeological stratigraphy, the final era of prehistoric time is then further divided into a three-age system: The Stone Age (2.1), the Bronze Age (2.2), and the Iron Age (2.3).
2.1 Stone age tools
2.2 Bronze age tools
2.3 Iron age tools
The Stone Age is the largest of the tripartite system, where humans primarily used stones as tools to make arrowheads, jewelry, small sculptures, and clothing. Lasting about 2.5 million years, the Stone Age ended around 9600 BCE at the advent of human skills in metalworking. Following the Stone Age, the Bronze Age followed characterized by the use of bronze, an alloy combination of tin and copper. Bronze allowed the beginnings of urban civilization, metal tools, government, and proto-writing. The third age, Iron, was the transition to smelting different types of metal to create iron and steel for weapons and tools for agriculture, ending around 600 BCE.
2.4 Rock outcrop
Prehistoric, indigenous cave people generally lived in sheltered places, hunted animals, and drew on rock walls. Despite being called cave people, most of them did not live in caves. Usually, the indigenous people lived in rock shelters or rock outcrops (2.4) with an overhanging roof for protection from the rain or snow but were open on at least two sides. Caves were dark and damp, animals hibernated in them, and caves were not always available. However, people did use caves that were safe or hidden away for multiple purposes (2.5), and in those hidden places, we have discovered a fantastic array of art.
2.5 Altamira bison
What were the prehistoric indigenous prehistoric people like? As explorers of the past, information is gathered based upon the archeological finds unearthed in the caves sealed for thousands of years. The artifacts give us a window to our distant relatives, and perhaps the biggest clue is found in their drawings and paintings left on the cave walls.


12.6: Exercises
Prove: If \(B\) is finite and \(A \subseteq B\text{,}\) then \(A\) is finite and \(\vert A \vert \le \vert B \vert\text{.}\)
Suppose that \(A\text{,}\) \(B\text{,}\) and \(C\) are finite subsets of a universal set \(U\text{.}\)
- Prove: If \(A\) and \(B\) are disjoint, then \(\vert A \sqcup B \vert = \vert A \vert + \vert B \vert\text{.}\)
- Prove: \(\vert A \cup B \vert = \vert A \vert + \vert B \vert - \vert A \cap B \vert\text{.}\)
- Hint.
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See Exercise 9.9.5 , and use the equality from Task a.
- Determine a similar formula for \(\vert A \cup B \cup C \text{.}\)
- Hint.
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Draw a Venn diagram first.
Use induction to prove directly that if \(\vert A \vert = n\) then \(\vert \mathscr{P}(A) \vert = 2^n\text{.}\) Use Worked Example 12.2.1 as a model for your proof of the induction step.
Prove: If \(\vert A \vert = \infty\) and \(A \subseteq B\text{,}\) then \(\vert B \vert = \infty\text{.}\)
Combine Example 12.3.3 and Example 12.3.10 to verify that the unit interval \((0,1)\) and \(\mathbb{R}\) have the same size.
- Hint.
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First map the punctured circle \(\hat{S}\) onto some open interval in the \(x\)-axis by “unrolling” \(\hat{S}\text{.}\)
Use Example 12.3.3 and the function \(f(x) = \tan x\) to prove that the interval \((-\pi/2,\pi/2)\) and \(\mathbb{R}\) have the same size.
- Hint.
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The function \(f(x) = \tan x\) is not one-to-one, but it becomes one-to-one if you restrict its domain to an appropriate interval
Prove that if \(A\) and \(B\) have the same size, then so do \(\mathscr{P}(A)\) and \(\mathscr{P}(B)\text{.}\)
- Hint.
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See Exercise 10.7.19 .
Suppose \(A\) is a set with \(\vert A \vert = n\text{.}\) Then we can enumerate its elements as \(A = \{a_1,a_2,\ldots ,a_n\}\text{.}\)
- Construct a bijection from the power set of \(A\) to the set of words in the alphabet \(\Sigma = \{T,F\}\) of length \(n\text{.}\)
Note that there are two tasks required here.
- Explicitly describe a function \(f: \mathscr{P}(A) \rightarrow \Sigma^\ast_n\) by describing the input-output rule: give a detailed description of how, given a subset \(B \subseteq A\text{,}\) the word \(f(B)\) should be produced.
- Prove that your function \(f\) is a bijection.
- Hint.
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When determining the input-output rule for your function \(f: \mathscr{P}(A) \rightarrow \Sigma^\ast_n\text{,}\) think of how one might construct an arbitrary subset of \(A\text{,}\) and then relate that process to a sequence of answers to \(n\) true/false questions.
- Use Task a to determine the cardinality of \(\mathscr{P}(A)\text{.}\) Explain.
- Hint
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See Note 1.3.1 .
- Suppose \(k\) is some fixed (but unknown) integer, with \(0 \le k \le n\text{.}\) Let \(\mathscr{P}(A)_k\) represent the subset of \(\mathscr{P}(A)\) consisting of all subsets of \(A\) that have exactly \(k\) elements. Describe how your bijection from Task a, could be used to count the elements of \(\mathscr{P}(A)_k\text{.}\)
- Hint.
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Consider how restricting the domain might help.