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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.

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 \(n + 1\).