General and overarching topics

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• ConceptMathematical proof

  A mathematical proof is an argument that shows that a statement follows from axioms, definitions and statements that are already proved. Each step of a proof follows from earlier steps by a valid rule of inference, such as modus ponens. A proved statement is a theorem.

• ConceptDirect proof

  A direct proof of an implication $p \to q$ assumes $p$ and then gets $q$ by a chain of valid steps. The steps use axioms, definitions and theorems that are already proved.

• ConceptProof by cases

  A proof by cases splits the situation into a finite number of cases that cover all possibilities. Then it proves the statement in each case. The basic form is this rule: from $p \lor q$, $p \to r$ and $q \to r$, conclude $r$.

• ConceptCounterexample

  A counterexample to a universal statement $\forall x \in X, \, P(x)$ is an element $a \in X$ such that $P(a)$ is false. One counterexample shows that the universal statement is false.

• ConceptProof by contradiction

  A proof by contradiction proves a proposition $p$ in two steps. It assumes $\neg p$, and it gets a contradiction from this assumption, that is, a statement $q$ together with $\neg q$. Then $\neg p$ is false, so $p$ is true by the double negation law.

• ConceptProof by contrapositive

  A proof by contrapositive proves an implication $p \to q$ by a direct proof of $\neg q \to \neg p$. The method is correct because $p \to q$ is logically equivalent to its contrapositive.