A proof is a valid argument that establishes the truth of a mathematical statement.

## Direct

A direct proof is structured to show \(p \rightarrow q\) based on the assumption that \(p\) is true.

Assume p, show q.

## Indirect

Not all statements may be expressed through direct establishedment that \(\forall x(P(x) \rightarrow Q(x))\). Indirect proof technique provide the tools for these cases.

### Contraposition

Contrapositives make use of \(p \rightarrow q \equiv \lnot q \rightarrow \lnot p\). In this case \(\lnot q\) is taken as a premise the it is shown that \(\lnot p\) must follow.

### Contradiction

To show \(p\) is true, demostrate that \(\lnot p \rightarrow (r \land \lnot r)\).

## Induction

Show that the statment holds in a base case where \(n = 0\) or \(1\).
Prove that for every \(n\) the statemetn holds for \(n + 1\).