Theorems

A theorem is a statement that can be shown to be true.

How are theorems stated?

• Many theorem assert that a property holds for all elements in the domain.
• We may have a universal quantifier, but it may not be explicitly stated.
  • Ex: If $x>y$, where $x$ and $y$ are positive real numbers, then $x^2>y^2$
  • What that really means is $(\forall x|x\in \mathbb{R}\wedge x>0:(\forall y|y\in \mathbb{R}\wedge y>0:x^2>y^2))$

Theorems that are Biconditional Statements

These are theorems that have the form $p\leftrightarrow q$.

We know that $p\leftrightarrow q\equiv (p\to q)\wedge (q\to p)$

We can prove biconditional by proving both $p\to q$ and $q\to p$.

Ex: For all integers $n$, $n$ is odd if and only if $n^2$ is odd.