Logic for the TMUA

This is a self-contained course to prepare students for the logic and proof topics tested in TMUA Paper 2. At the end of each section, there are worksheets with solutions to help students practice and reinforce their understanding. If you find any errors or have suggestions for improvement, please contact me via email at jzmaths@hotmail.com.

1. 1Basics of Implication and DeductionAn introduction to P => Q: what it means for one statement to imply another, and why 'deduced' and 'implied' are two words for the same idea.
2. 2P ⇒ Q: Equivalent Ways to Say ItThe many ways mathematicians express P ⇒ Q — if/then, only if, sufficient and necessary conditions — and why they all say the same thing.
3. 3Negation and QuantifiersHow to negate mathematical statements — including 'and'/'or' and quantifiers, simple and nested.
4. 4CounterexamplesWhat counterexamples are, how they disprove general claims, and how to disprove if-then statements by finding examples where the if part is true and the then part is false, including statements with quantifiers and nested implications.
5. 5Converse and EquivalenceThe converse of an implication and why it need not be true, and what happens when an implication and its converse both hold: equivalence, P if and only if Q, 'if and only if'.
6. 6The ContrapositiveHow to rewrite P => Q as not Q => not P, why this follows naturally from necessary conditions, and why the original implication and its contrapositive are equivalent.
7. 7Proof by ContradictionHow proof by contradiction works by assuming the opposite, reaching an impossible consequence, and connecting the method to the contrapositive.
8. 8Truth TablesTruth tables, why P ⇒ Q means 'P and not-Q can't both hold', vacuous truth, and a few other equivalent faces of the conditional.
9. 9TMUA Worked ExamplesThree TMUA-style logic questions with full worked solutions, pulling together implication, necessary and sufficient conditions, negation, quantifiers and the contrapositive.