• Proof
  • proof by induction
  • direct proof
  • contrapositive proof
  • proof by contradiction
  • circular proof

Proof

proof by induction

• variable
• property
• base case
• induction hypothesis
• induction step

direct proof

contrapositive proof

$P$ implies $Q$ is equivalent to $\sim Q$ implies $\sim P$

proof by contradiction

assume $\sim P$ is true, get a contradiction

circular proof

to prove $(1)\iff (2)\iff (3)$, need only prove $(1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (1)$