Undo It

One of the basic notions of functions that’s taught in Precalculus and in Discrete Mathematics is the notion of an inverse function: if $f: A \to B$ is a one-to-one and onto function, then there is an inverse function $f^{-1}: B \to A$ so that

$f^{-1}(f(a)) = a$ for all $a \in A$ and

$f(f^{-1}(b)) = b$ for all $b \in B$ .

If $A = B = \mathbb{R}$ , this is commonly taught in high school as a function that satisfies the horizontal line test.

In other words, if the function $f$ is applied to $a$ , the result is $f(a)$ . When the inverse function is applied to that, the answer is the original number $a$ . Therefore, I’ll tell my class, “By applying the function $f^{-1}$ , we uh-uh-uh-uh-uh-uh-uh-undo it.”

If I have a few country music fans in the class, this always generates a bit of a laugh.

See also the amazing duet with Carrie Underwood and Steven Tyler at the 2011 ACM awards:

Predicate Logic and Popular Culture (Part 31): The Godfather

Let $p$ be the proposition “I took the gun,” and let $q$ be the proposition “I took the cannoli.” Translate the logical statement

$\lnot p \land q$ .

Obviously, this is an allusion to one of the great lines in The Godfather.

Even though this is a simple example, it actually serves a pedagogical purpose (when I first introduce students to propositional logic) by illustrating two important points.

First, there is an order of precedence with $\lnot$ and $\land$ . Specifically, $\lnot p \land q$ means $(\lnot p) \land q$ (“I did not take the gun, and I took the cannoli”) and not $\lnot (p \land q)$ (“It is false that I took both the gun and the cannoli”).

Second, the actual line from The Godfather is not a proposition because both “Leave the gun” and “Take the cannoli” are commands. By contrast, a proposition must be a declarative sentence that is either true or false. That’s why I had to slightly modify the words to “I took the cannoli” instead of “Take the cannoli.”

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.

I’m afraid that I found plenty more examples from popular culture to illustrate predicate logic, but a month of posts on this topic is probably enough for now. I’ll return to this topic again at some point in the future.

Predicate Logic and Popular Culture (Part 30): The Platters

Let $R(x)$ be the proposition “ $x$ can make all this the world seem right,” and let $B(x)$ “ $x$ can make the darkness bright.” Translate the logical statement

$R(\hbox{you}) \land B(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot (R(x) \lor B(x)))$ ,

where the domain is all people.

The clunky way of translating this into English is, “You can make all this world seem right, you can make the darkness bright, and everyone else can neither make all this world seem right nor make the darkness bright.” Of course, this is the sentiment expressed by the first two lines of this classic by the Platters.

Predicate Logic and Popular Culture (Part 29): Grease

Let $W(x)$ be the proposition “I want $x$ .” Translate the logical statement

$W(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot W(x))$ ,

where the domain is all people.

The clunky way of translating this into English is, “I want you, and I don’t want anyone who isn’t you.” But it sounds a lot better when John Travolta and Olivia Newton-John sing it.

For professional mathematicians (as opposed to students first learning predicate logic), the more compact way of writing this would be

$W(\hbox{you}) \land \exists! x W(x)$ .

Predicate Logic and Popular Culture (Part 28): High School Musical

Let $L(x)$ be the proposition “ $x$ is a star in heaven” and let $R(x)$ be the proposition “We can reach $x$ ”

$\lnot \exists x(\lnot R(x))$ ,

where the domain for $x$ is the stars in heaven.

The clunky way of translating this into English is, “There is not a star in heaven that we cannot reach,” and this double negative appears in the song Breaking Free from High School Musical.

This example gives students a simple practice problem for using De Morgan’s laws to eliminate the double negative:

$\lnot \exists x(\lnot R(x)) \equiv \forall x(\lnot(\lnot R(x))) \equiv \forall x R(x)$ ,

or “We can reach every star in heaven.”

Predicate Logic and Popular Culture (Part 27): Les Miserables

Let $K(y)$ be the proposition “I know place $y$ ,” let $L(x,y)$ be the proposition “ $x$ is lost at place $y$ ,” and let $C(x,y)$ be the proposition “ $x$ cries at place $y$ .” Translate the logical statement

$\exists y(K(y) \land \lnot \exists x(L(x,y) \lor C(x,y)))$ ,

where the domain for $x$ is all people and the domain for $y$ is all places.

The clunky way of translating this into English is, “There exists a place that I know so that it is false that there is a person at this place who is lost or who cries.” This is the innocent childish dream of Cosette in Les Miserables as she suffers under the Thenardiers.

Predicate Logic and Popular Culture (Part 26): The Gatlin Brothers

Let $G(x)$ be the proposition “ $x$ is gold,” let $B(x)$ be the proposition “ $x$ is a bank,” and let $N(x,y)$ be the proposition “ $x$ is in $y$ .” Translate the logical statement

$\exists y(B(y) \land N(y, \hbox{the middle of Beverly Hills}) \land \forall x(G(x) \land N(x,\hbox{California}) \Longrightarrow N(x,y) \land N(x,\hbox{somebody else's name})))$

where the domain is all things.

Translating: “There is a bank in the middle of Beverly Hills so that all of the gold in California is in that bank and the gold is in someone else’s name. My father loved listening to country music, and I heard this hit of the 1970s repeatedly when I was a child.

Pedagogically, I like this example because it illustrates the subtle importance of the order of the quantifiers. Suppose I reversed the order:

$\forall x(G(x) \land N(x,\hbox{California}) \Longrightarrow N(x,\hbox{somebody else's name}) \land \exists y(B(y) \land N(x,y) \land N(y, \hbox{the middle of Beverly Hills})))$

The clunky way of translating this into English is, “All of the gold in California is in somebody else’s name, and for each piece of gold, there exists a bank such that the piece of gold is in the bank and the bank is in the middle of Beverly Hills.” That almost sounds like the first sentence, except that there is no guarantee that the same bank holds all of the gold. With this rendering, the song would be, “All the gold in California are in banks in the middle of Beverly Hills in somebody else’s name,” which is just a little bit different than what the Gatlin Brothers wrote.

I still remember, as a student, my professor impressing upon the order of the quantifier when I first learned about the notions of uniform continuity (as opposed to local continuity, the regular notion of continuity taught in calculus) and uniform convergence (as opposed to pointwise convergence).

Predicate Logic and Popular Culture (Part 25): Handel’s Messiah

Let $V(x)$ be the proposition “ $x$ is a valley,” and let $E(x,t)$ be the proposition “ $x$ is exalted at time $t$ .” Translate the logical statement

$\exists t \forall x (V(x) \Longrightarrow E(x,t))$ ,

where the domain for $x$ is all things and the domain for $t$ is all times.

The clunky way of translating this into English is, “There exists a time when every valley will be exalted at that time.” This provides the opening lyrics to the third part of Handel’s Messiah.

I’m personally partial to the modern rendition performed by Larnelle Harris.

In any event, Merry Christmas.

Predicate Logic and Popular Culture (Part 24): The Chipmunks

Let $G(t)$ be the proposition “We are good at time $t$ .” Translate the logical statement

$\forall t(x-1 \le t \le 0 \Longrightarrow H(t)) \land \exists t>0 \lnot H(t)$ ,

where the domain is all times (measured in years), time $0$ is now, and time $x$ is next Christmas.

The clunky way of translating this into English is, “We’ve been good at all times from last Christmas until now, but there will come a time when we are not good.” Of course, this is one of the lines in the famous Chipmunk Song.

Predicate Logic and Popular Culture (Part 23): My two front teeth

Let $W(x)$ be the proposition “I want $x$ for Christmas.” Translate the logical statement

$W(\hbox{my two front teeth}) \land \forall x (x \ne \hbox{my two front teeth} \Longrightarrow \lnot W(x))$ ,

where the domain is all things.

The clunky way of translating this into English is,”I want my two front teeth for Christmas, and if something isn’t my two front teeth, then I don’t want that for Christmas.”