Truth Tables and Equivalent Conditions

Boolean expressions built from the same variables can look completely different and still compute the same thing. The only way to know for sure is to check every possible input combination, because a condition that agrees with another on nine out of ten cases is not equivalent at all; it just hides its disagreement well. A truth table forces you to check all of them, systematically, so there's nowhere for a mismatch to hide.

To build one, list every variable once per column, then list every combination of true and false values as rows. With two variables you get four rows: TT, TF, FT, FF. With each variable you add, the table doubles in height, since every existing row now needs to pair with both a true and a false for the new variable. Three variables means eight rows, four variables means sixteen. For each row, plug the values into both expressions and work out the result using the usual rules: && is true only when both sides are true, || is true when at least one side is true, and ! flips a value. If the output columns for the two expressions match on every row, the expressions are equivalent, no matter how different they look written out.

This matters directly for writing conditions in code: an if statement's condition can often be rewritten in a cleaner or faster form, and a truth table is how you verify the rewrite didn't change behavior. It also connects to De Morgan's laws, which give shortcuts for rewriting negated && and || expressions without building a table from scratch every time.