Practice Direct Proof in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A direct proof establishes a statement P \Rightarrow Q by assuming P is true and using logical steps, definitions, and known theorems to arrive at Q — the most straightforward proof strategy.
Start from what you know (the hypotheses) and chain logical steps forward until you reach what you want to prove — no detours, no tricks, just forward reasoning.
Example 1
easyProve directly: the sum of an even integer and an odd integer is odd.
Example 2
mediumProve directly: for all real x, (x-1)(x+1) = x^2 - 1.
Example 3
easyProve directly: if x > y > 0, then x^2 > y^2.
Example 4
mediumProve directly: for all integers n, n^2 + n is even.