Nested conditionals show how equivalent statements may be written in different forms. Simplification of compound statements uses the laws of logic to find a shorter or cleaner equivalent form. This is a central skill in discrete mathematics because long formulas occur in proofs, circuits, program conditions, and switching networks. In this lesson, simplification of compound statements will be performed step by step with named equivalence laws.

Let $s$ be a compound statement. To \textbf{simplify} $s$ means to find a logically equivalent statement that uses fewer connectives, fewer negations, or fewer repeated components. Thus, if $s\Leftrightarrow t$ and $t$ is shorter or structurally simpler than $s$, then $t$ is a simplification of $s$.