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. Basics 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 very same idea.
  2. P ⇒ 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. Negation and Counter-ExamplesHow to negate mathematical statements — including 'and'/'or' and quantifiers, simple and nested — and how a single counter-example disproves a universal claim.
  4. Converse 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 ⇔ Q, 'if and only if'.
  5. The ContrapositiveWhy every implication P ⇒ Q can be read as not-Q ⇒ not-P, derived from what it means for Q to be necessary for P, plus the family of phrasings this opens up.
  6. Proof by ContradictionGrowing proof by contradiction out of the contrapositive: the method, the classic proof that √2 is irrational, and how contradiction differs from proof by contrapositive.
  7. Miscellaneous: Truth Tables and Loose EndsTruth tables, why P ⇒ Q means 'P and not-Q can't both hold', vacuous truth, and a few other equivalent faces of the conditional.
  8. TMUA Worked ExamplesThree TMUA-style logic questions with full worked solutions, pulling together implication, necessary and sufficient conditions, negation, quantifiers and the contrapositive.