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# Proofs

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.

To show $$p$$ is true, demostrate that $$\lnot p \rightarrow (r \land \lnot r)$$.
Show that the statment holds in a base case where $$n = 0$$ or $$1$$. Prove that for every $$n$$ the statemetn holds for $$n + 1$$.