Proof Techniques Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Compare direct proof and proof by contrapositive for: 'If n2n^2 is even, then nn is even.' Which technique is more natural here?

Solution

  1. 1
    Direct proof attempt: Assume n2n^2 is even, so n2=2kn^2 = 2k. We need to show nn is even. This is not immediate from n2=2kn^2 = 2k alone — harder.
  2. 2
    Contrapositive: assume nn is odd, so n=2m+1n = 2m+1. Then n2=(2m+1)2=4m2+4m+1=2(2m2+2m)+1n^2 = (2m+1)^2 = 4m^2+4m+1 = 2(2m^2+2m)+1, which is odd. So n2n^2 is not even.
  3. 3
    Since the contrapositive is easier to prove (odd assumption gives a concrete form), it is more natural here.

Answer

Contrapositive is more natural: assume n odd, show n2 odd\text{Contrapositive is more natural: assume } n \text{ odd, show } n^2 \text{ odd}
The contrapositive transforms 'if n2n^2 even then nn even' into 'if nn odd then n2n^2 odd' — the hypothesis nn odd gives n=2m+1n = 2m+1, a concrete form to work with. Direct proof of the original is harder because 'n2=2kn^2 = 2k' is less tractable.

About Proof Techniques

Proof techniques are standard strategies for establishing mathematical claims under different structures.

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More Proof Techniques Examples

Example 1 easy

Name four proof techniques, give a one-sentence description of each, and identify which is best suit

Example 3 easy

Which proof technique is most appropriate for: 'There exists a real number [formula] such that [form

Example 4 medium

Prove using mathematical induction: [formula] for all [formula].

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