Q&A: What’s the Intuition for Why If-Then and Only-If are Logically Equivalent?

by Justin Skycak on

Cross-posted from here.

Question

I teach high school math. Some of my colleagues insist that a proof by induction should explicitly refer to the principle of mathematical induction, i.e. it must include the words “by the principle of mathematical induction” or words of equivalent meaning.

Others say that (given a proposition $H_n$) it is enough to show that the base case is true and that $H_k\implies H_{k+1}$; it is not necessary to explicitly refer to the principle of mathematical induction.

Answer

The appropriate level of granularity for a proof depends on the audience.

  • If you're taking an "Intro to Proofs" class and your homework is to do some proofs by induction, then yeah, you will be expected to state when, where, and how you use the principle of mathematical induction. You should explicitly label the base case, the inductive hypothesis, etc. Not because your grader needs that information to follow your proof, but because the whole point of the homework is to demonstrate that you know this stuff. (Likewise, if you're teaching an "Intro to Proofs" class then obviously you need to label all those steps in your example proofs to provide proper scaffolding for students.)
  • If you're a full-blown mathematician who is writing a research paper, then your audience already knows this stuff like the back of their hand, so most of it doesn't need to be said explicitly. Sure, it will make the reading a little gentler if you precede the inductive proof with "By induction, ...", and that would probably be a nice thing to do for your audience, but you're not going to lose any other research mathematician in your audience if you omit the explicit reference to induction, and nobody is going to question your understanding of induction because you didn't explicitly label something (you'd have to make a legitimate and egregious error in the basic logic for that to happen).


Your case sounds closer to the first one, so I would say sure, as long as your colleagues tell their students what features of the induction proof they are required to explicitly state/label, it’s appropriate for them to expect a student’s induction proof to include those features.

… though if they’re reading a math research publication and claiming that it should be edited to include the include the words “by the principle of mathematical induction”, then that’s a different story ;)