Practice question · Select all that apply
Select every statement that is TRUE about mathematical induction.
Hints
- Induction climbs a ladder of discrete rungs.
- Ask whether the reals can be reached one step at a time.
Show the answer
- A. It requires a base case
- B. It requires an inductive step
- D. The inductive hypothesis assumes the claim for one value k
- E. It proves infinitely many cases with a finite argument
Why
Induction needs both obligations, assumes the claim at a single k, and settles infinitely many cases finitely. But it works only over well-ordered discrete sets like the naturals, the reals have no 'next' number, so there is no rung to step to.
Practise Mathematical Induction
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More questions on Mathematical Induction
- An inductive step alone, without a base case, is enough to prove a statement for all n.
- Order the parts of a proof by induction that 1 + 2 + ... + n = n(n+1)/2.
- Using the formula 1 + 2 + ... + n = n(n+1)/2, compute the sum of the first 10 natural numbers.
- Complete the statement of the inductive step.
- An induction proof assumes the very statement it is trying to prove holds for n, then proves it for n+1. Why…
- A famous fake proof shows all horses are the same colour by induction: true for one horse, and any n+1 horses…