My Favorite One-Liners: Part 89

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.

Here’s a problem that might arise in my discrete mathematics class:

Find the negation of p \Rightarrow q.

This requires a couple of reasonably complex steps. First, we use the fact that p \Rightarrow q is logically equivalent to $\lnot p \lor q$:

\lnot(p \Rightarrow q) \equiv \lnot (\lnot p \lor q).

Next, we have to apply DeMorgan’s Law to find the negation:

\lnot (p \Rightarrow q) \equiv \lnot(\lnot p \lor q) \equiv \lnot(\lnot p) \land \lnot q

Finally, we arrive at the final step: simplifying \lnot(\lnot p). At this point, I tell my class, it’s a bit of joke, especially after the previous, more complicated steps. “Not not p,” of course, is the same as p. So this step is a bit of a joke. Which steps up the following cringe-worthy pun:

In fact, you might even call this a not-not joke.

After the groans settle down, we finish the derivation:

\lnot(p \Rightarrow q) \equiv \lnot(\lnot p \lor q) \equiv \lnot(\lnot p) \land \lnot q \equiv p \land \lnot q.

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