Maths (university) · Part 01
Logic, sets and proof
The language and proof methods every university maths course relies on.
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- StatementA sentence that is definitely true or definitely false, never both. Maths proofs are built from statements.
- Implication'If P then Q'. It is false only when P is true and Q is false. In every other case it counts as true.
- ContrapositiveThe implication 'if not Q then not P'. It is true exactly when 'if P then Q' is true, so you can prove either one.
- Quantifiers'For all' says a claim holds for every member of a set. 'There exists' says it holds for at least one. Negating swaps one into the other.
- Direct proofStart from the assumptions and move step by step, using definitions and known facts, until you reach the conclusion.
- Proof by contradictionAssume the opposite of what you want to prove. Follow the logic until something impossible appears. So the opposite was wrong.
- InductionTo prove a claim for every whole number n, show it for n = 1 (base case). Then show that truth for n = k forces truth for n = k + 1.
- Set operationsA set is a collection of distinct objects. The union joins two sets, the intersection keeps what they share, and A minus B removes B's members from A.
- Injective and surjectiveAn injective function never sends two different inputs to the same output. A surjective function hits every value in its target set. Both together is a bijection.