Truth table

A truth table is a table that lists all possible states of a statement. Truth tables are commonly used to compare statements; if two statements share the same truth table, then the two statements are said to be logically equivalent. Truth tables are also able to be used to find negations of statements.

Equivalent statements
If two statements, $$P$$ and $$Q$$, have truth tables that contain the same elements, then the two statements are logically equivalent. That is, the two statements will be true or false under the same conditions. $$P$$ and $$Q$$ being logically equivalent is denoted as: $$P\equiv Q$$. Finding equivalent statements is a tool used in mathematics as it may be difficult to directly prove one statement, but easy to prove an equivalent statement.

Example of equivalent statements
One can show that, for two statements $$P$$ and $$Q$$, $$P\implies Q$$ is logically equivalent to $$\neg P\or Q$$ by showing that they have the same truth table. Consider the truth table for $$P\implies Q$$:

In the above table, T is true and F is false. Now consider the truth table for $$\neg P\or Q$$:

Observing the two tables, they have the same elements. Thus, $$P\implies Q$$ is logically equivalent to $$\neg P\or Q$$.

Negation of statements
Given a statement $$P$$, another statement $$Q$$ is the negation of $$P$$ if the state of $$Q$$ is opposite of $$P$$ given the same conditions. That is, $$Q$$ is false when $$P$$ is true, and $$Q$$ is true when $$P$$ is false. If this holds, then $$Q$$ is the negation of $$P$$, written as $$\neg P$$ (sometimes as ~$$P$$).

The negation of $$P$$, $$\neg P$$, is visualized in the above truth table.

Example of negated statements
One can show that $$P\and\neg Q$$ is the negation of $$P\implies Q$$ via a truth table.