# Definition:Truth Function/Connective

## Definition

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:

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