# 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, $\displaystyle P$ and $\displaystyle Q$ , 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. $\displaystyle P$ and $\displaystyle Q$ being logically equivalent is denoted as: $\displaystyle P\equiv Q$ . 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 $\displaystyle P$ and $\displaystyle Q$ , $\displaystyle P\implies Q$ is logically equivalent to $\displaystyle \neg P\or Q$ by showing that they have the same truth table. Consider the truth table for $\displaystyle P\implies Q$ :

$\displaystyle P\implies Q$
$\displaystyle P$ $\displaystyle Q$ $\displaystyle P\implies Q$
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 $\displaystyle \neg P\or Q$ :

$\displaystyle \neg P\or Q$
$\displaystyle P$ $\displaystyle Q$ $\displaystyle \neg P\or Q$
T T T
F F T
T F F
F T T

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

## Negation of statements

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

$\displaystyle P$ $\displaystyle \neg P$
T F
F T

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

### Example of negated statements

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

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