Discover published sets by community

Proof by induction is used primarily to prove statements about integers. It consists of two steps: the base case where the statement is proved for an initial value (usually the smallest integer in question), and the inductive step where one proves that if the statement holds for an arbitrary integer $n$, it must also hold for $n+1$.

Direct proof entails showing that a given statement logically follows from the axioms of the system using a sequence of logical deductions. It is often used to prove implications (if-then statements) such as $p \rightarrow q$ by assuming $p$ and showing $q$ must follow.

In proof by contradiction, also known as reductio ad absurdum, one assumes the opposite of the statement to be proven and shows that this assumption leads to a contradiction. Since the contradiction implies that the assumption is false, the original statement must be true.

Proof by exhaustion, or proof by cases, involves dividing a problem into a finite number of cases and proving that the statement is true for each case. It is frequently used when there are a small, manageable number of scenarios to consider.