My Favorite One-Liners: Part 12

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.

Often in mathematics, one proof is quite similar to another proof. For example, in Precalculus or Discrete Mathematics, students encounter the theorem

$\sum_{k=1}^n (a_k + b_k) = \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$ .

The formal proof requires mathematical induction, but the “good enough” proof is usually convincing enough for most students, as it’s just the repeated use of the commutative and associative properties to rearrange the terms in the sum:

$\sum_{k=1}^n (a_k + b_k)= (a_1 + b_1) + (a_2 + b_2) + \dots + (a_n + b_n)$

$= (a_1 + a_2 + \dots + a_n) + (b_1 + b_2 + \dots + b_n)$

$= \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$ .

Next, I’ll often present the new but closely related theorem

$\sum_{k=1}^n (a_k - b_k) = \sum_{k=1}^n a_k -\sum_{k=1}^n b_k$ .

The proof of this would take roughly the same amount of time as the first proof, but there’s often little pedagogical value in doing all the steps over again in class. So here’s the line I’ll use: “At this point, I invoke the second-most powerful word in mathematics…” and then let them guess what this mysterious word is.

After a few seconds, I tell them the answer: “Similar.” The proof of the second theorem exactly parallels the proof of the first except for some sign changes. So I’ll tell them that mathematicians often use this word in mathematical proofs when it’s dead obvious that the proof can be virtually copied-and-pasted from a previous proof.

Eventually, students will catch on to my deliberate choice of words and ask, “What the most powerful word in mathematics?” As any mathematician knows, the most powerful word in mathematics is “Trivial”… the proof is so easy that it’s not necessary to write the proof down. But I warn my students that they’re not allowed to use this word when answering exam questions.

The third most powerful phrase in mathematics is “It is left for the student,” thus saving the professor from writing down the proof in class and encouraging students to figure out the details on their own.

My Favorite One Liners: Part 2

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 doing a large computation, I’ll often leave plenty of blank space on the board to fill it later. For example, when proving by mathematical induction that

$1 + 3 + 5 + \dots + (2n-1) = n^2$ ,

the inductive step looks something like

$1 + 3 + 5 \dots + (2k-1) + (2[k+1]-1) =$

$~$

$= (k+1)^2$

So I explained that, to complete the proof by induction, all we had to do was convert the top line into the bottom line.

As my class swallowed hard as they thought about how to perform this task, I told them, “Yes, this looks really intimidating. Indeed, to quote the great philosopher, ‘You might think that I’m insane. But I’ve got a blank space, baby… so let’s write what remains.’ “

And, just in case you’ve been buried under a rock, here’s the source material for the one-liner (which, at the time of this writing, is the fifth-most watched video on YouTube):

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

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