Induction can refer to weak induction, strong induction, or structural induction. In all cases, induction is a method for proving a statement about a "complex" element of a set by reducing it to a "simpler" case.
To prove "weak induction, you must prove (this is often called the base case), and then you must prove for an arbitrary , assuming (this is called the inductive step)." using
To prove weak induction, you can prove and prove for an arbitrary , assuming .by
This is just a change of variables, but it occasionally makes the notation a bit easier to work with.
There are other variants that you can use. For example, if you only care about for , you can use the following principle:
To prove weak induction, you can prove and prove for an arbitrary , assuming .using
This is also equivalent to weak induction.
You can mix and match these variations. If you're using an unfamiliar variation, you should check that it makes sense intuitively, and if possible, show how to convert it to a proof by weak induction. For example, you should be sure to avoid a backwards proof while doing induction.
Strong inductionStrong induction is similar to weak induction, except that you make additional assumptions in the inductive step.
- prove base case), and (this is called the
- for an arbitrary , prove , assuming (this is the inductive step)
More concisely, the inductive step requires you to prove for all .assuming
The intuition for why strong induction works is the same reason as that for weak induction: in order to prove , for example, I would first use the base case to conclude . Next, I would use the inductive step to prove ; this inductive step may use but that's ok, because we've already proved . I would then use the inductive step to conclude ; this may use both and , but that's okay because we've already proved and . Next, I would again use the inductive step to conclude ; as before, this may use , , or , but this is not a problem since we have already proved those three facts. Similarly, we can use the inductive step to conclude P(4), P(5), etc.
Note that you can always use strong induction instead of weak induction. Using weak induction is just a matter of style: by avoiding unneeded assumptions, you reduce the complexity of your proof, and clearly indicate to the reader what assumptions you are actually planning to use. I often start inductive proofs by not specifying whether they are proofs by strong or weak induction; once I know which inductive hypothesis I actually need, I go back and fill in the beginning of my inductive step.