Proof by Cases

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Questions?

Test Question 2

A sketch of the logic for showing that if n is even, then n³ is also even. A full-credit answer on the test would also phrase this logic as a formal proof.

Assume n is even, i.e., n = 2a for some integer a. Then n³ = (2a)³ = 8a³ = 2(4a³). By closure under multiplication, 4a³ is an integer and so n³, which is twice that integer, is even.

Proof In Cases

Section 3.4

Warm-Up Example

Define f(x) = x² if x < 0, f(x) = 2x if x ≥ 0. Prove that f(x) is non-negative for all real numbers x.

The proof can be done in cases, with each case corresponding to one part of the function definition. This is a natural way to use proof in cases with piecewise function definitions.

Here is the formal proof (Theorem 1 in the linked document, and its proof), and its LaTeX source.

Another Example

Theorem: every integer multiple of 5 is either of the form 10n or 10n + 5, for some integer n.

This too can be proved by cases, although the cases don’t come quite as directly from the theorem statement.

Here’s the formal proof (Theorem 2 in the linked document) and its LaTeX source.

Next

Congruence and the division algorithm.

Read section 3.5.

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