My Favorite One-Liners: Part 106

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

$\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}}$ ,

$\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}}$ ,

$\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}$ .

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

My Favorite One-Liners: Part 74

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

After presenting the Fundamental Theorem of Calculus to my calculus students, I make a point of doing the following example in class:

$\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx$

Hopefully my students are able to produce the correct answer:

$\displaystyle \int_0^4 \frac{1}{4} x^2 \, dx = \displaystyle \left[ \frac{x^3}{12} \right]^4_0$

$= \displaystyle \frac{(4)^3}{12} - \frac{(0)^3}{12}$

$= \displaystyle \frac{64}{12}$

$= \displaystyle \frac{16}{3}$

Then I tell my students that they’ve probably known the solution of this one since they were kids… and I show them the classic video “Unpack Your Adjectives” from Schoolhouse Rock. They’ll watch this video with no small amount of confusion (“How is this possibly connected to calculus?”)… until I reach the 1:15 mark of the video below, when I’ll pause and discuss this children’s cartoon. This never fails to get an enthusiastic response from my students.

If you have no idea what I’m talking about, be sure to watch the first 75 seconds of the video below. I think you’ll be amused.

Predicate Logic and Popular Culture (Part 123): Willie Nelson

Let $M(t)$ be the proposition “You were on my mind at time $t$ .” Translate the logical statement

$\forall t < 0 (M(t))$ .

Naturally, this matches the classic song by Willie Nelson (though Elvis did record it before him).

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 122): Queen

Let $p$ be the proposition “I cross a million rivers,” let $q$ be the proposition “I rode a million miles,” and let $r$ be the proposition “I still am where I started.” Translate the logical statement

$(p \land q) \Rightarrow r$ .

This matches a line from this classic by Queen.

Predicate Logic and Popular Culture (Part 121): OneRepublic

Let $F(x)$ be the proposition “ $x$ is a right friend,” let $P(y)$ be the proposition “ $y$ is a right place,” let $I(x,y)$ be the proposition “ $x$ is located at place $y$ ,” and let $H(x,y)$ be the proposition “They have $x$ at place $y$ ,” and let $p$ be the proposition “We’re going down.” Translate the logical statement

$\forall x \forall y(F(x) \land P(y) \land I(x,y) \Rightarrow H(x,y)) \land p$ .

This matches the chorus of this song by OneRepublic.

Predicate Logic and Popular Culture (Part 120): Crossfade

Let $C(t)$ be the proposition “At time $t$ , I meant to be so cold.” Translate the logical statement

$\forall t < 0 \lnot C(t)$ .

This matches the echo of this song by Crossfade.

Predicate Logic and Popular Culture (Part 119): Billy Joel

Let $p$ be the proposition “I’m gonna try for an uptown girl,” let $B(x)$ the proposition “ $x$ has hot blood,” let $q$ be the proposition “She’s looking for a downtown man,” and let $r$ be the proposition “I’m a downtown man.” Also, define the function $f(x)$ to be how long $x$ has lived in a white bread world. Translate the logical statement

$p \land \forall x (B(x) \Rightarrow (f(x) \le f(\hbox{she})) \land q \land r$ .

Of course, this matches the first chorus of the Billy Joel classic.

Predicate Logic and Popular Culture (Part 118): Bruno Mars

Let $D(x)$ be the proposition “Today I am doing $x$ .” Translate the logical statement

$\forall x \lnot D(x)$ .

This matches the closing line of the chorus of the Bruno Mars song.

Predicate Logic and Popular Culture (Part 117): Kelly Clarkson

Let $K(x)$ be the proposition “ $x$ kills you,” let $S(x)$ be the proposition “ $x$ makes you stronger,” and let $T(x)$ be the proposition “ $x$ makes you stand a little taller.” Translate the logical statement

$\forall x( \lnot K(x) \Rightarrow (S(x) \land T(x)))$ .

This matches the first line of this hit song by Kelly Clarkson.

Predicate Logic and Popular Culture (Part 116): George Strait

Let $G(x)$ be the proposition “I’ve got $x$ ,” let $P(x)$ be the proposition “ $x$ is ocean-front property,” and let $A(x)$ be the proposition “ $x$ is in Arizona.” Translate the logical statement

$\exists x (G(x) \land P(x) \land A(x))$ .