My Favorite One-Liners: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on my favorite one-liners.

Mathematical Wisecracks for Almost Any Occasion: Part 2, Part 7, Part 8, Part 12, Part 21, Part 28, Part 29, Part 41, Part 46, Part 53, Part 60, Part 63, Part 65, Part 71, Part 79, Part 84, Part 85, Part 100, Part 101, Part 108, Part 109, Part 114, Part 118, Part 121

All-Purpose Anecdotes: Part 38, Part 50, Part 64, Part 70, Part 92, Part 94

Addressing Misconceptions: Part 3, Part 4, Part 11, Part 14, Part 15, Part 18, Part 30, Part 32, Part 33, Part 37, Part 45, Part 59

Tricky Steps in a Calculation: Part 5, Part 6

Greek alphabet and choice of variables: Part 40, Part 43, Part 56

Homework and exams: Part 39, Part 47, Part 55, Part 57, Part 58, Part 66, Part 77, Part 78, Part 91, Part 96, Part 97, Part 107

Inequalities: Part 99

Simplification: Part 10, Part 102, Part 103

Polynomials: Part 19, Part 48, Part 49, Part 81, Part 90

Inverses: Part 16

Exponential and Logarithmic Functions: Part 1, Part 42, Part 68, Part 80, Part 110

Trigonometry: Part 9, Part 69, Part 76, Part 106, Part 120

Complex numbers: Part 54, Part 67, Part 86, Part 112, Part 113

Sequences and Series: Part 20, Part 35, Part 111

Combinatorics: Part 27

Statistics: Part 22, Part 23, Part 36, Part 51, Part 52, Part 61, Part 95, Part 116

Probability: Part 26, Part 31, Part 62, Part 93, Part 122

Calculus: Part 24, Part 25, Part 72, Part 73, Part 74, Part 75, Part 83, Part 87, Part 88, Part 104, Part 115, Part 117

Logic and Proofs: Part 13, Part 17, Part 34, Part 44, Part 89, Part 98, Part 119

Differential Equations: Part 82, Part 105

Confirming Einstein’s General Theory of Relativity with Calculus: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on general relativity and the precession of Mercury’s orbit.

Part 1: Introduction

Part 1a: Introduction
Part 1b: Observed precession of Mercury
Part 1c: Outline of argument

Part 2: Precession, polar coordinates, and conic sections

Part 2a: Graphically exploring precession
Part 2b: Polar coordinates and ellipses
Part 2c: Polar coordinates, circles, and parabolas
Part 2d: Polar coordinates and hyperbolas

Part 3: Method of successive approximations

Part 4: Principles from physics

Part 4a: Angular momentum
Part 4b: Acceleration in polar coordinates
Part 4c: Newton’s Second Law and Newton’s Law of Gravitation

Part 5: Orbits under Newtonian mechanics

Part 5a: Confirmation of solution
Part 5b: Derivation with calculus
Part 5c: Derivation with differential equations and the method of undetermined coefficients
Part 5d: Derivation with differential equations and variation of parameters

Part 6: Orbits under general relativity

Part 6a: New differential equation under general relativity
Part 6b: Confirmation of solution
Part 6c: Derivation with variation of parameters
Parts 6d, 6e, 6f, 6g, 6h, 6i, 6j: Rationale for the method of undetermined coefficients
Part 6k: Derivation with undetermined coefficients

Part 7: Computing precession

Parts 7a, 7b, 7c, 7d: Predicting precession
Part 7e: Computing precession

Part 8: Second- and third-order solutions with the method of successive approximations

Part 9: Pedagogical thoughts

Earlier this year, I presented these ideas for the UNT Math Department’s Undergraduate Mathematics Colloquium Series. The video of my lecture is below.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 9: Mentally computing $n$ given $n^5$ if $10 \le n \le 99$ .

Part 10: A mathematical optical illusion.

Part 11: The 27-card trick, which requires representing numbers in base 3.

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

Lagrange Points and Polynomial Equations: Part 5

This series was motivated by a terrific article that I read in the American Mathematical Monthly about Lagrange points, which are (from Wikipedia) “points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies.” There are five such points in the Sun-Earth system, called $L_1$ , $L_2$ , $L_3$ , $L_4$ , and $L_5$ .

The article points out a delicious historical factoid: Lagrange had a slight careless mistake in his derivation!

From the article:

Equation (d) would be just the tool to use to determine where to locate the JWST [James Webb Space Telescope, which is now in orbit about $L_2$ ], except for one thing: Lagrange got it wrong!… Do you see it? His algebra in converting $1 - \displaystyle \frac{1}{(m-1)^3}$ to common denominator form is incorrect… Fortunately, at some point in the two-and-a-half centuries between Lagrange’s work and the launch of JWST, this error has been recognized and corrected.

This little historical anecdote illustrates that, despite our best efforts, even the best of us are susceptible to careless mistakes. The simplification should have been

$q' = \displaystyle \left[ 1 - \frac{1}{(m-1)^3} \right] \cdot \frac{1}{r^3}$

$= \displaystyle \frac{(m-1)^3 - 1}{(m-1)^3} \cdot \frac{1}{r^3}$

$= \displaystyle \frac{m^3 - 3m^2 + 3m - 1 - 1}{(m-1)^3} \cdot \frac{1}{r^3}$

$= \displaystyle \frac{m^3 - 3m^2 + 3m - 2}{(m-1)^3} \cdot \frac{1}{r^3}$ .

(Parenthetically, The article also notes a clear but unintended typesetting error, as the correct but smudged exponent of 3 in the first equation became an incorrect exponent of 2 in the second.)

Lagrange Points and Polynomial Equations: Part 4

From Wikipedia, Lagrange points are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. There are five such points in the Sun-Earth system, called $L_1$ , $L_2$ , $L_3$ , $L_4$ , and $L_5$ .

The stable equilibrium points $L_4$ and $L_5$ are easiest to explain: they are the corners of equilateral triangles in the plane of Earth’s orbit. The points $L_1$ and $L_2$ are also equilibrium points, but they are unstable. Nevertheless, they have practical applications for spaceflight.

As we’ve seen, the positions of $L_1$ and $L_2$ can be found by numerically solving the fifth-order polynomial equations

$t^5 - (3-\mu) t^4 + (3-2\mu)t^3 - \mu t^2 + 2\mu t - \mu = 0$

and

$t^5 + (3-\mu) t^4 + (3-2\mu)t^3 - \mu t^2 - 2\mu t - \mu = 0$ ,

respectively. In these equations, $\mu = \displaystyle \frac{m_2}{m_1+m_2}$ where $m_1$ is the mass of the Sun and $m_2$ is the mass of Earth. Also, $t$ is the distance from the Earth to $L_1$ or $L_2$ measured as a proportion of the distance from the Sun to Earth.

We’ve also seen that, for the Sun and Earth, $mu \approx 3.00346 \times 10^{-6}$ , and numerically solving the above quintics yields $t \approx 0.000997$ for $L_1$ and $t \approx 0.01004$ for $L_2$ . In other words, $L_1$ and $L_2$ are approximately the same distance from Earth but in opposite directions.

There’s a good reason why the positive real roots of these two similar quintics are almost equal. We know that $t$ will be a lot closer to 0 than 1 because, for gravity to balance, the Lagrange points have to be a lot closer to Earth than the Sun. For this reason, the terms $\mu t^2$ and $2\mu t$ will be a lot smaller than $\mu$ , and so those two terms can be safely ignored in a first-order approximation. Also, the terms $t^5$ and $(3-\mu)t^4$ will be a lot smaller than $(3-2\mu)t^3$ , and so those two terms can also be safely ignored in a first-order approximation. Furthermore, since $\mu$ is also close to 0, the coefficient $(3-2\mu)$ can be safely replaced by just $3$ .

Consequently, the solution of both quintic equations should be close to the solution of the cubic equation

$3t^3 - \mu = 0$ ,

which is straightforward to solve:

$3t^3 = \mu$

$t^3 = \displaystyle \frac{\mu}{3}$

$t = \displaystyle \sqrt[3]{ \frac{\mu}{3} }$ .

If $\mu = 3.00346 \times 10^{-6}$ , we obtain $t \approx 0.010004$ , which is indeed reasonably close to the actual solutions for $L_1$ and $L_2$ . Indeed, this may be used as the first approximation in Newton’s method to quickly numerically evaluate the actual solutions of the two quintic polynomials.

Happy Pythagoras Day!

Let me take a break from my current series of posts to wish everyone an early Happy Pythagoras Day! Tomorrow will be 10/26/24 (or 26/10/24 in other parts of the world), and $26^2 = 10^2 + 24^2$ .

Bonus points if you can figure out (without Googling) when the next Pythagoras Day will be.

Lagrange Points and Polynomial Equations: Part 3

The position of $L_2$ can be found by numerically solving the fifth-order polynomial equation

$(m_1+m_2)t^5+(3m_1+2m_2)t^4+(3m_1+m_2)t^3$

$-(m_2+3m_3)t^2-(2m_2+3m_3)t-(m_2+m_3)=0$ .

In this equation, $m_1$ is the mass of the Sun, $m_2$ is the mass of Earth, $m_3$ is the mass of the spacecraft, and $t$ is the distance from the Earth to $L_2$ measured as a proportion of the distance from the Sun to Earth. In other words, if the distance from the Sun to Earth is 1 unit, then the distance from the Earth to $L_2$ is $t$ units. The above equation is derived using principles from physics which are not elaborated upon here.

We notice that the coefficients of $t^5$ , $t^4$ , and $t^3$ are all positive, while the coefficients of $t^2$ , $t$ , and the constant term are all negative. Therefore, since there is only one change in sign, this equation has only one positive real root by Descartes’ Rule of Signs.

Since $m_3$ is orders of magnitude smaller than both $m_1$ and $m_2$ , this may safely approximated by

$(m_1+m_2)t^5+(3m_1+2m_2)t^4+(3m_1+m_2)t^3 - m_2 t^2- 2m_2 t-m_2=0$ .

This new equation can be rewritten as

$t^5 + \displaystyle \frac{3m_1 + 2m_2}{m_1 + m_2} t^4 + \frac{3m_1+m_2}{m_1+m_2} t^3 - \frac{m_2}{m_1+m_2} t^2 - \frac{2m_2}{m_1+m_2} t- \frac{m_2}{m_1+m_2} = 0$ .

If we define

$\mu = \displaystyle \frac{m_2}{m_1+m_2}$ ,

we see that

$\displaystyle \frac{3m_1 + 2m_2}{m_1 + m_2} = \frac{3m_1 + 3m_2}{m_1 + m_2} - \frac{m_2}{m_1 + m_2} = 3 - \mu$

and

$\displaystyle \frac{3m_1 + m_2}{m_1 + m_2} = \frac{3m_1 + 3m_2}{m_1 + m_2} - \frac{2m_2}{m_1 + m_2} = 3 - 2\mu$ ,

so that the equation may be written as

$t^5 + (3-\mu) t^4 + (3-2\mu) - \mu t^2 - 2\mu t - \mu = 0$ ,

matching the equation found at Wikipedia.

For the Sun and Earth, $m_1 \approx 1.9885 \times 10^{30} ~ \hbox{kg}$ and $m_2 \approx 5.9724 \times 10^{24} ~ \hbox{kg}$ , so that

$mu = \displaystyle \frac{5.9724 \times 10^{24}}{1.9885 \times 10^{30} + 5.9724 \times 10^{24}} \approx 3.00346 \times 10^{-6}$ .

This yields a quintic equation that is hopeless to solve using standard techniques from Precalculus, but the root can be found graphically by seeing where the function crosses the $x-$ axis (or, in this case, the $t-$ axis):

As it turns out, the root is $t \approx 0.01004$ , so that $L_2$ is located $1.004\%$ of the distance from the Earth to the Sun in the direction away from the Sun.

Lagrange Points and Polynomial Equations: Part 2

We begin with $L_1$ , whose position can be found by numerically solving the fifth-order polynomial equation

$(m_1+m_3)x^5+(3m_1+2m_3)x^4+(3m_1+m_3)x^3$

$-(3m_2+m_3)x^2-(3m_2+m_3)x-(m_2+m_3)=0$ .

In this equation, $m_1$ is the mass of the Sun, $m_2$ is the mass of Earth, $m_3$ is the mass of the spacecraft, and $x$ is the distance from the Earth to $L_1$ measured as a proportion of the distance from the Sun to $L_1$ . In other words, if the distance from the Sun to $L_1$ is 1 unit, then the distance from the Earth to $L_1$ is $x$ units. The above equation is derived using principles from physics which are not elaborated upon here.

We notice that the coefficients of $x^5$ , $x^4$ , and $x^3$ are all positive, while the coefficients of $x^2$ , $x$ , and the constant term are all negative. Therefore, since there is only one change in sign, this equation has only one positive real root by Descartes’ Rule of Signs.

Since $m_3$ is orders of magnitude smaller than both $m_1$ and $m_2$ , this may safely approximated by

$m_1 x^5 + 3m_1 x^4 + 3m_1 x^3 - 3m_2 x^2 - 3m_2x - m_2=0$ .

Unfortunately, the unit $x$ is not as natural for Earth-bound observers as $t$ , the proportion of the distance of $L_1$ to Earth as a proportion of the distance from the Sun to Earth. Since $L_1$ is between the Sun and Earth, the distance from the Sun to Earth is $x+1$ units, so that $t = x/(x+1)$ . We then solve for $x$ in terms of $t$ (just like finding an inverse function):

$t = \displaystyle \frac{x}{x+1}$

$t(x+1) = x$

$tx + t = x$

$t = x - tx$

$t= x(1-t)$

$\displaystyle \frac{t}{1-t} = x$ .

Substituting into the above equation, we find an equation for $t$ :

$\displaystyle \frac{m_1t^5}{(1-t)^5} + \frac{3m_1t^4}{(1-t)^4} + \frac{3m_1t^3}{(1-t)^3} - \frac{3m_2t^2}{(1-t)^2} - \frac{3m_2t}{1-t} - m_2=0$

$m_1t^5 + 3m_1t^4(1-t) + 3m_1t^3(1-t)^2 - 3m_2t^2(1-t)^3 - 3m_2t(1-t)^4 - m_2(1-t)^5=0$

Expanding, we find

$m_1 t^5 + 3m_1 (t^4 - t^5) + 3m_1 (t^3-2t^4+t^5) - 3m_2 (t^2-3t^3+3t^4-t^5)$

$-3m_2(t - 4t^2 + 6t^3 - 4t^4 + t^5) - m_2(1 - 5t + 10t^2 - 10 t^3 + 5t^4 + t^5) = 0$

Collecting like terms, we find

$(m_1 - 3m_1 + 3m_1 + 3m_2 - 3m_2 + m_2)t^5 + (3m_1-6m_1-9m_2+12m_2-5m_2)t^4$

$+ (3m_1+9m_2-18m_2+10m_2)t^3 + (-3m_2+12m_2-10m_2) t^2$

$+ (-3m_2+5m_2)t - m_2 = 0$ ,

$(m_1+m_2) t^5 - (3m_1 +2m_2) t^4 + (3m_1 + m_2) t^3 - m_2 t^2 + 2m_2 t- m_2 = 0$ .

Again, this equation has only one positive real root since the original quintic in $x$ only had one positive real root. This new equation can be rewritten as

$t^5 - \displaystyle \frac{3m_1 + 2m_2}{m_1 + m_2} t^4 + \frac{3m_1+m_2}{m_1+m_2} t^3 - \frac{m_2}{m_1+m_2} t^2 + \frac{2m_2}{m_1+m_2} t- \frac{m_2}{m_1+m_2} = 0$ .

If we define

$\mu = \displaystyle \frac{m_2}{m_1+m_2}$ ,

we see that

$\displaystyle \frac{3m_1 + 2m_2}{m_1 + m_2} = \frac{3m_1 + 3m_2}{m_1 + m_2} - \frac{m_2}{m_1 + m_2} = 3 - \mu$

and

$\displaystyle \frac{3m_1 + m_2}{m_1 + m_2} = \frac{3m_1 + 3m_2}{m_1 + m_2} - \frac{2m_2}{m_1 + m_2} = 3 - 2\mu$ ,

so that the equation may be written as

$t^5 + (\mu-3) t^4 + (3-2\mu) - \mu t^2 + 2\mu t - \mu = 0$ ,

matching the equation found at Wikipedia.

For the Sun and Earth, $m_1 \approx 1.9885 \times 10^{30} ~ \hbox{kg}$ and $m_2 \approx 5.9724 \times 10^{24} ~ \hbox{kg}$ , so that

$\mu = \displaystyle \frac{5.9724 \times 10^{24}}{1.9885 \times 10^{30} + 5.9724 \times 10^{24}} \approx 3.00346 \times 10^{-6}$ .

As it turns out, the root is $t \approx 0.00997$ , so that $L_1$ is located $0.997\%$ of the distance from the Earth to the Sun in the direction of the Sun.

Lagrange Points and Polynomial Equations: Part 1

I recently read a terrific article in the American Mathematical Monthly about Lagrange points, which are (from Wikipedia) “points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies.” There are five such points in the Sun-Earth system, called $L_1$ , $L_2$ , $L_3$ , $L_4$ , and $L_5$ .

To describe these Lagrange points, I can do no better than the estimable Isaac Asimov. I quote from his essay “Colonizing the Heavens” from his book The Beginning and the End, which was published in 1977. I read the book over and over again as a boy in the mid-1980s. (Asimov’s essay originally concerned the Earth-Moon system; in the quote below, I changed the words to apply to the Sun-Earth system.)

Imagine the Sun at zenith, exactly overhead. Trace a line due eastward from the Sun down to the horizon. Two-thirds of the way along that line, one-third of the way up from the horizon, is one of those places. Trace another line westward away from the Sun down to the horizon. Two-thirds of the way along that line, one-third of the way up from the horizon, is another of those places.

Put an object in either place and it will form an equilateral triangle with the Sun and Earth…

What is so special about those places? Back in 1772, the astronomer Joseph Louis Lagrange showed that in those places any object remained stationary with respect to the Sun. As the Earth moved about the Sun, any object in either of those places would also move about the Sun in such a way as to keep perfect step with the Earth. The competing gravities of the Sun and Earth would keep it where it was. If anything happened to push it out of place it would promptly move back, wobbling back and forth a bit (“librating”) as it did so. The two places are called “Lagrangian points” or “libration points.”

Lagrange discovered five such places altogether, but three of them are of no importance since they don’t represent stable conditions. An object in those three places, once pushed out of place, would continue to drift outward and would never return.

The last paragraph of the above quote represents a rare failure of imagination by Asimov, who wrote prolifically about the future of spaceflight. Points $L_4$ and $L_5$ are indeed stable equilibria, and untold science fiction stories have placed spacecraft or colonies at these locations. (The rest of Asimov’s essay speculates about using these points in the Earth-Moon system for space colonization.) However, while the points $L_1$ and $L_2$ are unstable equilibria, they do have practical applications for spacecraft that can perform minor course corrections to stay in position. (The point $L_3$ is especially unstable to outside gravitational influences and thus seems unsuitable for spacecraft.) Again from Wikipedia,

Sun–Earth L₁ is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere… Solar and heliospheric missions currently located around L₁ include the Solar and Heliospheric Observatory, Wind, Aditya-L1 Mission and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe(IMAP) and the NEO Surveyor.

Sun–Earth L₂ is a good spot for space-based observatories. Because an object around L₂ will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler… The James Webb Space Telescope was positioned in a halo orbit about L₂ on January 24, 2022.

Earth–Moon L₁ allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable space station intended to help transport cargo and personnel to the Moon and back. The SMART-1 Mission passed through the L₁ Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon’s gravitational influence.

Earth–Moon L₂ has been used for a communications satellite covering the Moon’s far side, for example, Queqiao, launched in 2018, and would be “an ideal location” for a propellant depot as part of the proposed depot-based space transportation architecture.

While the locations $L_4$ and $L_5$ are easy to describe, the precise locations of $L_1$ and $L_2$ are found by numerically solving a fifth-order polynomial equation. This was news to me when I read that article from the American Mathematical Monthly. While I had read years ago that finding the positions of the other three Lagrange points wasn’t simple, I did not realize that it was no more complicated that numerically finding the roots of a polynomial.

The above article from the American Mathematical Monthly concludes…

[t]he mathematical tools that Lagrange uses to arrive at a solution to this three-body problem lie entirely within the scope of modern courses in algebra, trigonometry, and first-semester calculus. But surely no ordinary person could have pursued the many extraordinarily complicated threads in his work to their ends, let alone woven them together into a magnificent solution to the problem as he has done. Lagrange noted in the introduction to his essay, “This research is really no more than for pure curiosity …” If only he could have watched on Christmas Day as the James Webb Space Telescope began its journey to the Lagrange point $L_2$ !

In this short series, we discuss the polynomial equations for finding $L_1$ and $L_2$ .

Predicate Logic and Popular Culture: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on using examples from popular culture to illustrate principles of predicate logic. My experiences teaching these ideas to my discrete mathematics students led to my recent publication (John Quintanilla, “Name That Tune: Teaching Predicate Logic with Popular Culture,” MAA Focus, Vol. 36, No. 4, pp. 27-28, August/September 2016).

Unlike other series that I’ve made, this series didn’t have a natural chronological order. So I’ll list these by concept illustrated from popular logic. The vast majority of these examples were suggested by my students.

Logical and $\land$ :

Part 1: “You Belong To Me,” by Taylor Swift
Part 21: “Do You Hear What I Hear,” covered by Whitney Houston
Part 31: The Godfather (1972)
Part 45: The Blues Brothers (1980)
Part 53: “What Does The Fox Say,” by Ylvis
Part 54: “Billie Jean,” by Michael Jackson
Part 98: “Call Me Maybe,” by Carly Rae Jepsen.
Part 137: “Pen Pineapple Apple Pen,” by Pikotaro.
Part 142: “Galway Girl,” by Ed Sheeran.
Part 183: A memorable line from Avengers: Age of Ultron.
Part 184: A memorable line from Star Wars Episode I: The Phantom Menace.
Part 197: “That’s Life,” by Frank Sinatra.
Part 209: A line from The Office.
Part 242: A line from The Fellowship of the Ring.
Part 249: “We Didn’t Start the Fire,” by Billy Joel.
Part 252: A line from The Two Towers.
Part 253: “I’m Gonna Miss Her,” by Brad Paisley.
Part 261: A line from Spiderman 2.

Logical or $\lor$ :

Part 1: Shawshank Redemption (1994)

Logical negation $\lnot$ :

Part 1: Richard Nixon
Part 32: “Satisfaction!”, by the Rolling Stones
Part 39: “We Are Never Ever Getting Back Together,” by Taylor Swift
Part 129: “Blue Ain’t Your Color,” by Keith Urban.
Part 143: “Ain’t Worth The Whiskey,” by Cole Swindell.
Part 247: A line from The Fellowship of the Ring.

Logical implication $\Rightarrow$ :

Part 1: Field of Dreams (1989), and also “Roam,” by the B-52s
Part 2: “Word Crimes,” by Weird Al Yankovic
Part 7: “I’ll Be There For You,” by The Rembrandts (Theme Song from Friends)
Part 43: “Kiss,” by Prince
Part 50: “I’m Still A Guy,” by Brad Paisley
Part 76: “You’re Never Fully Dressed Without A Smile,” from Annie.
Part 109: “Dancing in the Dark,” by Bruce Springsteen.
Part 122: “Keep Yourself Alive,” by Queen.
Part 140: “It Don’t Mean A Thing If It Ain’t Got That Swing,” by Ella Fitzgerald.
Part 174: A famous line from Rocky IV.
Part 176: A famous line from Game of Thrones.
Part 185: A line from Westworld.
Part 188: A line from Talladega Nights.
Part 195: “If We Were a Movie,” from the Hannah Montana series.
Part 207: A line from Name of the Wind, by Patrick Rothfuss.
Part 222: A line from The Notebook.
Part 234: “Battle Symphony,” by Linkin Park.
Part 235: A line from Suits.
Part 259: “Out There,” from “The Hunchback of Notre Dame.”
Part 269: “Dear Theodosia,” from Hamilton.

For all $\forall$ :

Part 3: Casablanca (1942)
Part 4: A Streetcar Named Desire (1951)
Part 34: “California Girls,” by The Beach Boys
Part 37: Fellowship of the Ring, by J. R. R. Tolkien
Part 49: “Buy Me A Boat,” by Chris Janson
Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)
Part 65: “Stars and Stripes Forever,” by John Philip Sousa.
Part 68: “Love Yourself,” by Justin Bieber.
Part 69: “I Will Always Love You,” by Dolly Parton (covered by Whitney Houston).
Part 74: “Faithfully,” by Journey.
Part 79: “We’re Not Gonna Take It Anymore,” by Twisted Sister.
Part 87: “Hungry Heart,” by Bruce Springsteen.
Part 99: “It’s the End of the World,” by R.E.M.
Part 100: “Hold the Line,” by Toto.
Part 101: “Break My Stride,” by Matthew Wilder.
Part 102: “Try Everything,” by Shakira.
Part 108: “BO$$,” by Fifth Harmony.
Part 113: “Sweet Caroline,” by Neil Diamond.
Part 114: “You Know Nothing, Jon Snow,” from Game of Thrones.
Part 118: “The Lazy Song,” by Bruno Mars.
Part 120: “Cold,” by Crossfade.
Part 123: “Always on My Mind,” by Willie Nelson.
Part 127: Motif from Hamilton.
Part 131: Abraham Lincoln’s Second Inaugural Address.
Part 132: A famous line from The Fellowship of the Ring.
Part 133: A famous line from Braveheart.
Part 136: “I Don’t Wanna Live Forever,” by ZAYN and Taylor Swift.
Part 138: “Bohemian Rhapsody,” by Queen.
Part 144: “The Anchor,” by Bastille.
Part 146: A line from the video game series “Fallout”
Part 147: “Nobody’s Perfect,” from the Hannah Montana series.
Part 150: “Roar,” by Katy Perry.
Part 167: “Look What You Made Me Do,” by Taylor Swift.
Part 180: “Bohemian Rhapsody,” by Queen.
Part 181: “We Don’t Talk Anymore,” by Charlie Puth and Selena Gomez.
Part 189: “Tengo Muchas Alas / I Have Many Wings,” by Mana.
Part 190: “Eastside,” by Benny Blanco, Halsey, and Khalid.
Part 191: “I’m a Mess,” by Bebe Rexha.
Part 193: “Forever and Ever, Amen,” by Randy Travis.
Part 202: “Everything Is Awesome!!!,” from The Lego Movie.
Part 204: “She’s Always A Woman,” by Billy Joel.
Part 208: A line by Naruto in Masashi Kishimoto’s anime.
Part 211: “Everyone Lies To Me,” by Knuckle Puck.
Part 214: “Aeroplane,” by Björk.
Part 216: “Hound Dog,” by Elvis Presley.
Part 227: One Fish, Two Fish, Red Fish, Blue Fish, by Dr. Seuss.
Part 230: “Be Alright,” by Dean Lewis.
Part 231: “Everyone Wants To Be A Cat,” from The Aristocats.
Part 232: “Behind Blue Eyes,” by Limp Bizkit.
Part 236: A line from Dirty Dancing.
Part 243: “Better,” by Khalid.
Part 246: “Unfaithful,” by Rihanna.
Part 260: A line from Ratatouille.
Part 263: A line from Shakespeare’s As You Like It.
Part 264: “Bet On It,” from High School Musical 2.
Part 271: A line from Pirates of the Caribbean: The Curse of the Black Pearl.
Part 273: “Gaston,” from Beauty and the Beast.

For all and implication:

Part 8 and Part 9: “What Makes You Beautiful,” by One Direction
Part 13: “Safety Dance,” by Men Without Hats
Part 16: The Fellowship of the Ring, by J. R. R. Tolkien
Part 24 : “The Chipmunk Song,” by The Chipmunks
Part 55: The Quiet Man (1952)
Part 62: “All My Exes Live In Texas,” by George Strait.
Part 70: “Wannabe,” by the Spice Girls.
Part 72: “You Shook Me All Night Long,” by AC/DC.
Part 81: “Ascot Gavotte,” from My Fair Lady
Part 82: “Sharp Dressed Man,” by ZZ Top.
Part 86: “I Could Have Danced All Night,” from My Fair Lady.
Part 95: “Every Breath You Take,” by The Police.
Part 96: “Only the Lonely,” by Roy Orbison.
Part 97: “I Still Haven’t Found What I’m Looking For,” by U2.
Part 105: “Every Rose Has Its Thorn,” by Poison.
Part 107: “Party in the U.S.A.,” by Miley Cyrus.
Part 112: “Winners Aren’t Losers,” by Donald J. Trump and Jimmy Kimmel.
Part 115: “Every Time We Touch,” by Cascada.
Part 117: “Stronger,” by Kelly Clarkson.
Part 125: “Do Wot You Do,” by INXS.
Part 130: “Think of You,” by Chris Young and Cassadee Pope.
Part 135: “Can’t Feel My Face,” by The Weeknd.
Part 145: A line from Black Dynamite.
Part 152: “You Haven’t Done Nothin’,” by Stevie Wonder.
Part 155: “All The Lazy Boyfriends,” by They Might Be Giants.
Part 165: A famous quote by Eleanor Roosevelt.
Part 166: “Perfect,” by Ed Sheeran.
Part 172: “Twas the Night Before Christmas,” by Clement Clarke Moore.
Part 182: “How Far I’ll Go,” from Moana.
Part 192: A line from the videogame “Overwatch.”
Part 194: A line from the Dragon Ball franchise.
Part 196: “ME!,” by Taylor Swift.
Part 200: A line from the 1990s Spider-Man cartoons.
Part 201: “It’s Quiet Uptown,” from Hamilton.
Part 205: “Three Little Birds,” by Bob Marley.
Part 206: “Mudfootball,” by Jack Johnson.
Part 210: “On My Way,” by Alan Walker, Sabrina Carpenter, and Farruko.
Part 215: “Aging Rockers,” by Tim Hawkins.
Part 218: “Happy Together,” by The Turtles.
Part 223: “Too Deep to Turn Back,” by Daniel Caesar.
Part 224: “Nothing Gold Can Stay,” by Robert Frost.
Part 225: “He Stopped Loving Her Today,” by George Jones.
Part 229: A line from Mean Girls.
Part 233: “Emperor’s New Clothes,” by Panic! At The Disco.
Part 240: “For Those About To Rock (We Salute You),” by AC/DC.
Part 244: A famous line from Spiderman.
Part 248: The poem “All That Is Gold Does Not Glitter,” from The Fellowship of the Ring.
Part 251: “Survey Ladies,” from Animaniacs.
Part 265: “Every Time You Go Away,” by Paul Young.
Part 268: “Sweet Dreams,” by Eurhymics.
Part 270: A line from the anime “Naruto Shippuden.”
Part 272: “Be Our Guest,” from Beauty and the Beast.
Part 274: “He Stopped Loving Her Today,” by George Jones.

There exists $\exists$ :

Part 10: “Unanswered Prayers,” by Garth Brooks
Part 15: “Stand by Your Man,” by Tammy Wynette (also from The Blues Brothers)
Part 36: Hamlet, by William Shakespeare
Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)
Part 93: “There’s No Business Like Show Business,” from Annie Get Your Gun (1946).
Part 94: “Not While I’m Around,” from Sweeney Todd (1979).
Part 104: “Wild Blue Yonder” (US Air Force)
Part 106: “No One,” by Alicia Keys.
Part 116: “Ocean Front Property,” by George Strait.
Part 139: “Someone in the Crowd,” from La La Land.
Part 149: “Someone Like You,” by Adele.
Part 151: “E-MO-TION,” by Carly Rae Jepsen.
Part 154: “I Wanna Dance With Somebody,” by Whitney Houston.
Part 162: “Think of You,” by Chris Young and Cassadee Pope.
Part 168: “Sorry,” by Halsey.
Part 175: “Someday We’ll Be Together,” by Diana Ross and the Supremes.
Part 177: “Try Everything,” by Shakira.
Part 186: “Someday,” by Nickelback.
Part 226: “The Wizard and I,” from Wicked.
Part 238: A line from the video game “Among Us.”
Part 250: “Ain’t No Mountain High Enough,” by Marvin Gaye.

Existence and uniqueness:

Part 14: “Girls Just Want To Have Fun,” by Cyndi Lauper
Part 20: “All I Want for Christmas Is You,” by Mariah Carey
Part 23: “All I Want for Christmas Is My Two Front Teeth,” covered by The Chipmunks
Part 29: “You’re The One That I Want,” from Grease
Part 30: “Only You,” by The Platters
Part 35: “Hound Dog,” by Elvis Presley
Part 73: “Dust In The Wind,” by Kansas.
Part 75: “Happy Together,” by The Turtles.
Part 77: “All She Wants To Do Is Dance,” by Don Henley.
Part 90: “All You Need Is Love,” by The Beatles.
Part 169: “Marry Me,” by Thomas Rhett.
Part 179: A line from “Harry Potter and the Sorcerer’s Stone.”
Part 245: An advertising line for Gibson guitars.
Part 258: “Nobody Knows,” by Kevin Sharp.
Part 262: “Dust in the Wind,” by Kansas.

DeMorgan’s Laws:

Part 5: “Never Gonna Give You Up,” by Rick Astley
Part 28: “We’re Breaking Free,” from High School Musical (2006)
Part 255: A line from “The Nightmare Before Christmas.”
Part 257: A line from “The Wizard of Oz.”

Simple nested predicates:

Part 6: “Everybody Loves Somebody Sometime,” by Dean Martin
Part 25: “Every Valley Shall Be Exalted,” from Handel’s Messiah
Part 33: “Heartache Tonight,” by The Eagles
Part 38: “Everybody Needs Somebody To Love,” by Wilson Pickett and covered in The Blues Brothers (1980)
Part 46: “Mean,” by Taylor Swift
Part 56: “Turn! Turn! Turn!” by The Byrds
Part 63: P. T. Barnum.
Part 64: Abraham Lincoln.
Part 66: “Somewhere,” from West Side Story.
Part 71: “Hold On,” by Wilson Philips.
Part 80: Liverpool FC.
Part 84: “If You Leave,” by OMD.
Part 103: “The Caisson Song” (US Army).
Part 111: “Always Something There To Remind Me,” by Naked Eyes.
Part 121: “All the Right Moves,” by OneRepublic.
Part 126: Motif from Hamilton.
Part 157: “Whenever, Wherever,” by Shakira.
Part 158: “Church Bells,” by Carrie Underwood.
Part 163: A famous line from The Princess Bride.
Part 170: “Everywhere,” by Tim McGraw.
Part 173: “If I Ain’t Got You,” by Alicia Keys.
Part 187: “Always Something There To Remind Me,” by Naked Eyes.
Part 198: “All Star,” by Smash Mouth.
Part 203: “Lean On Me,” by Bill Withers.
Part 217: A line in the video game Valorant.
Part 219: “Señorita,” by Shawn Mendes and Camila Cabello.
Part 220: “How to Love,” by Cash Cash.
Part 221: A line from Monk.
Part 237: A line from Psycho.
Part 254: “You Can’t Always Get What You Want,” by the Rolling Stones.
Part 256: “Irgendwie, Irgendwo, Irgendwann” by Nena.
Part 266: “Kokomo,” by the Beach Boys.

Maximum or minimum of a function:

Part 12: “For the First Time in Forever,” by Kristen Bell and Idina Menzel and from Frozen (2013)
Part 19: “Tennessee Christmas,” by Amy Grant
Part 22: “The Most Wonderful Time of the Year,” by Andy Williams
Part 48: “I Got The Boy,” by Jana Kramer
Part 60: “I Loved Her First,” by Heartland
Part 92: “Anything You Can Do,” from Annie Get Your Gun.
Part 119: “Uptown Girl,” by Billy Joel.
Part 124: “All I Want To Do Is Be With You,” from High School Musical 3.
Part 160: “God, Your Mama, and Me,” by Florida Georgia Line and the Backstreet Boys.
Part 178: “Ex Factor,” by Lauryn Hill.

Somewhat complicated examples:

Part 11 : “Friends in Low Places,” by Garth Brooks
Part 27 : “There is a Castle on a Cloud,” from Les Miserables
Part 41: Winston Churchill
Part 44: Casablanca (1942)
Part 51: “Everybody Wants to Rule the World,” by Tears For Fears
Part 58: “Fifteen,” by Taylor Swift
Part 59: “We Are Never Ever Getting Back Together,” by Taylor Swift
Part 61: “Style,” by Taylor Swift
Part 67: “When I Think Of You,” by Janet Jackson.
Part 78: “Nothing’s Gonna Stop Us Now,” by Starship.
Part 89: “No One Is Alone,” from Into The Woods.
Part 110: “Everybody Loves My Baby,” by Louis Armstrong.
Part 134: A famous line from Braveheart.
Part 141: “How Far I’ll Go,” from Moana.
Part 148: “The Climb,” by Miley Cyrus.
Part 153: “I Can’t Tell You Why,” by The Eagles.
Part 161: “For What It’s Worth,” by Buffalo Springfield.
Part 164: “When The Sun Goes Down,” by Kenny Chesney.
Part 199: “Never Say Never,” by Justin Bieber.
Part 213: “Sign of the Times,” by Harry Styles.
Part 241: “Dreams,” by Fleetwood Mac.
Part 267: A line from The Hound of the Baskervilles.

Fairly complicated examples:

Part 17 : Richard Nixon
Part 47: “Homegrown,” by Zac Brown Band
Part 52: “If Ever You’re In My Arms Again,” by Peabo Bryson
Part 83: “Something Good,” from The Sound of Music.
Part 85: “Joy To The World,” by Three Dog Night.
Part 88: “Like A Rolling Stone,” by Bob Dylan.
Part 91: “Into the Fire,” from The Scarlet Pimpernel.
Part 128: “A Puzzlement,” from The King and I.
Part 156: “Everybody Loves a Lover,” by Doris Day.
Part 159: “Fastest Girl in Town,” by Miranda Lambert.
Part 171: “Everybody’s Got Somebody But Me,” by Hunter Hayes.

Really complicated examples:

Part 18: “Sleigh Ride,” covered by Pentatonix
Part 26: “All the Gold in California,” by the Gatlin Brothers
Part 40: “One of These Things Is Not Like the Others,” from Sesame Street
Part 42: “Take It Easy,” by The Eagles