My Favorite One-Liners: Part 44

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.

Today’s quip is something that I’ll use to emphasize that the meaning of the word “or” is a little different in mathematics than in ordinary speech. For example, in mathematics, we could solve a quadratic equation for $x$ :

$x^2 + 2x - 8 = 0$

$(x+4)(x-2) = 0$

$x + 4 = 0 \qquad \hbox{OR} \qquad x - 2 = 0$

$x = -4 \qquad \hbox{OR} \qquad x = 2$

In this example, the word “or” means “one or the other or maybe both.” It could be that both statements are true, as in the next example:

$x^2 + 2x +1 = 0$

$(x+1)(x+1) = 0$

$x + 1 = 0 \qquad \hbox{OR} \qquad x + 1= 0$

$x = -1 \qquad \hbox{OR} \qquad x = -1$

However, in plain speech, the word “or” typically means “one or the other, but not both.” Here the quip I’ll use to illustrate this:

At the end of “The Bachelor,” the guy has to choose one girl or the other. He can’t choose both.

My Favorite One-Liners: Part 38

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.

When I was a student, I heard the story (probably apocryphal) about the mathematician who wrote up a mathematical paper that was hundreds of pages long and gave it to the departmental administrative assistant to type. (This story took place many years ago before the advent of office computers, and so typewriters were the standard for professional communication.) The mathematician had written “iff” as the standard abbreviation for “if and only if” since typewriters did not have a button for the $\Leftrightarrow$ symbol.

Well, so the story goes, the administrative assistant saw all of these “iff”s, muttered to herself about how mathematicians don’t know how to spell, and replaced every “iff” in the paper with “if”.

And so the mathematician had to carefully pore through this huge paper, carefully checking if the word “if” should be “if” or “iff”.

I have no idea if this story is true or not, but it makes a great story to tell students.

My Favorite One-Liners: Part 34

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.

Suppose that my students need to prove a theorem like “Let $n$ be an integer. Then $n$ is odd if and only if $n^2$ is odd.” I’ll ask my students, “What is the structure of this proof?”

The key is the phrase “if and only if”. So this theorem requires two proofs:

Assume that $n$ is odd, and show that $n^2$ is odd.
Assume that $n^2$ is odd, and show that $n$ is odd.

I call this a blue-light special: Two for the price of one. Then we get down to the business of proving both directions of the theorem.

I’ll also use the phrase “blue-light special” to refer to the conclusion of the conjugate root theorem: if a polynomial $f$ with real coefficients has a complex root $z$ , then $\overline{z}$ is also a root. It’s a blue-light special: two for the price of one.

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

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