Level 7: How a proof works

Logic and puzzles · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

In the earlier levels you learned the tools of logic: statements, truth tables, the contrapositive, quantifiers and valid arguments. In this level you use them to do what mathematicians do every day: prove things, so that they are known to be true for every case, forever. You meet the main methods (direct proof, contrapositive, contradiction, cases and counterexamples), learn to write a proof in clear sentences, and end with the paradoxes that show what happens when logic is used carelessly.

Lesson	You will be able to
What a proof is	Tell evidence from proof, and name the definitions, axioms and earlier theorems a proof stands on
Direct proof	Prove facts about odd, even and divisible numbers straight from the definitions, written in full sentences
Proof by contrapositive	Prove "if \(n^2\) is even then \(n\) is even", and decide when the contrapositive is the easier road
Proof by contradiction	Prove that there is no largest even number and that \(\sqrt2\) is irrational, and say exactly what a contradiction is
Counterexamples and cases	Break a false claim with one counterexample, and prove a true one by splitting it into cases
Paradoxes	Explain the liar, the barber and the surprise test, and say what each one teaches about logic

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