Analyzing a Simple Proof

Intro to Proofs · Axiom Academy

EXAMPLE Analyzing a Simple Proof Dissect a proof line by line — identifying the hypothesis, the definitions, the algebra, and the conclusion. The sum of two even integers is even. Proof Structure & Key Takeaways Nice work — you dissected the whole proof. Every proof of this kind has the same three-part shape: We start by clearly stating what we are given: m and n are even integers. This is the starting point. We apply the definition of "even," perform algebraic manipulations (substitution, factoring), and use a property of the integers (closure under addition). Each step follows logically from the one before. We show that m + n has the form 2k for an integer k , which is exactly the definition of even. That proves what we set out to prove. Definitions are fundamental: we used the definition of "even" twice — once to unpack what we were given, and again to conclude what we proved. Every step must be justified: each line follows from a definition, a previous step, or a known property of the integers. Algebra serves a purpose: factoring out the 2 was not arbitrary — it revealed the 2(a+b) structure needed to apply the definition of "even." The conclusion must match the goal: we transformed m + n into the form 2k , which is precisely what "even" requires. As you read and write harder proofs, keep asking: What definitions apply? What properties can I use? How does each step connect to the next? That analytical habit is what proof-reading is all about.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.