# Definition:Truth Function

## Definition

Let $\mathbb B$ be the set of truth values, and let $k$ be a natural number.

A mapping $f: \mathbb B^k \to \mathbb B$ is called a truth function.

### Truth Functions of Connectives

The logical connectives are assumed to be truth-functional.

Hence, they are represented by certain truth functions.

#### Logical Negation

The logical not connective defines the truth function $f^\neg$ as follows:

 $\displaystyle \map {f^\neg} \F$ $=$ $\displaystyle \T$ $\displaystyle \map {f^\neg} \T$ $=$ $\displaystyle \F$

#### Logical Conjunction

The conjunction connective defines the truth function $f^\land$ as follows:

 $\displaystyle \map {f^\land} {F, F}$ $=$ $\displaystyle F$ $\displaystyle \map {f^\land} {F, T}$ $=$ $\displaystyle F$ $\displaystyle \map {f^\land} {T, F}$ $=$ $\displaystyle F$ $\displaystyle \map {f^\land} {T, T}$ $=$ $\displaystyle T$

#### Logical Disjunction

The disjunction connective defines the truth function $f^\lor$ as follows:

 $\displaystyle f^\lor \left({F, F}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\lor \left({F, T}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\lor \left({T, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\lor \left({T, T}\right)$ $=$ $\displaystyle T$

#### Conditional

The conditional connective defines the truth function $f^\to$ as follows:

 $\displaystyle f^\to \left({F, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\to \left({F, T}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\to \left({T, F}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\to \left({T, T}\right)$ $=$ $\displaystyle T$

#### Biconditional

The biconditional connective defines the truth function $f^\leftrightarrow$ as follows:

 $\displaystyle f^\leftrightarrow \left({F, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\leftrightarrow \left({F, T}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\leftrightarrow \left({T, F}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\leftrightarrow \left({T, T}\right)$ $=$ $\displaystyle T$

#### Exclusive Disjunction

The exclusive or connective defines the truth function $f^\oplus$ as follows:

 $\displaystyle f^\oplus \left({F, F}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\oplus \left({F, T}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\oplus \left({T, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\oplus \left({T, T}\right)$ $=$ $\displaystyle F$

#### Logical NAND

The NAND connective defines the truth function $f^\uparrow$ as follows:

 $\displaystyle f^\uparrow \left({F, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\uparrow \left({F, T}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\uparrow \left({T, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\uparrow \left({T, T}\right)$ $=$ $\displaystyle F$

#### Logical NOR

The NOR connective defines the truth function $f^\downarrow$ as follows:

 $\displaystyle f^\downarrow \left({F, F}\right)$ $=$ $\displaystyle T$ $\displaystyle f^\downarrow \left({F, T}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\downarrow \left({T, F}\right)$ $=$ $\displaystyle F$ $\displaystyle f^\downarrow \left({T, T}\right)$ $=$ $\displaystyle F$

## Also known as

Some sources hyphenate: truth-function.

Others speak of a boolean function or a boolean operator, alluding to the fact that there are two possible outputs.

## Also see

• Results about truth functions can be found here.