# Truth table

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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 $P\implies Q$: $P\implies Q$ $P$ $Q$ $P\implies Q$
T T T
F F T
T F F
F T T

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

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 ~ $P$). $P$ $\neg P$
T F
F T

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. $P$ $Q$ $P\implies Q$ $P\and\neg Q$
T T T F
F F T F
T F F T
F T T F