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A2 • Proof • 10 minutes

Proof by Deduction

Proof by deduction starts with known facts or definitions and uses valid algebraic steps to reach a conclusion that must be true.

Deductive proofEven numbersAnimated algebraPractice included
🎬 Animated Lesson: Deduction
a=2mb=2n2(m+n)∎

Worked Example

Question

Prove that the sum of two even numbers is even.

Strategy

Use two different variables because the two even numbers do not have to be the same. Let them be 2m and 2n.

Final Written Proof

Let a and b be even integers. Then a = 2m and b = 2n for some integers m and n. So: a + b = 2m + 2n = 2(m + n) Since m + n is an integer, a + b is two times an integer. Therefore, the sum of two even numbers is even. ∎

Quick Check

Why do we use m and n instead of using the same letter twice?

🤖 AI Tutor Placeholder

Later this can let students ask: “Can you check my proof?” or “Give me a harder deduction question”.