About this topic
Before you begin
Write direct, contrapositive, contradiction, case, induction, existence and uniqueness proofs, and use the pigeonhole and symmetry principles and combinatorial arguments.
Learning goals
- Construct a direct proof from definitions
- Disprove a universal statement with a verified counterexample
- Prove a statement by contradiction or by its contrapositive
- Prove a statement by an exhaustive case analysis
- Prove statements by ordinary or strong mathematical induction
- Prove existence, uniqueness or a construction
- Apply the pigeonhole principle
- Prove an identity or count by a combinatorial argument
- Use symmetry to reduce or complete an argument
Syllabus statements covered
- Include: Direct proof; Disproof by counterexample; Proof by contradiction; Proof of existence; Proof of uniqueness; Proof by construction; Proof by cases; Proof by mathematical induction; Pigeonhole principle; Symmetry principle; Combinatorial arguments and proofs.
Choose a lesson
Work through them in order.
- Not startedContrapositive, contradiction, cases and counterexamplesNextChoose a proof method from the structure of a claim, write the proof in full and disprove false claims with a checked counterexample.
- Not startedProof by mathematical inductionNextProve statements about every integer from a starting value, use strong induction when one earlier case is not enough, and find gaps in flawed inductions.
- Not startedExistence, uniqueness and constructionNextProve that an object exists, by construction or without one, and prove that it is the only one.
- Not startedThe pigeonhole principle and combinatorial proofsNextChoose boxes for pigeonhole arguments and prove identities by counting the same set in two ways.
- Not startedThe symmetry principleNextUse symmetry to evaluate integrals, reduce cases and find probabilities without counting.