# Negation of Compound Statements

## Negation of Conjunction:

A conjunction p ^ q consists of two sub-statements p and q both of which exist simultaneously. Therefore, the negation of the conjunction would mean the negation of at least one of the two sub-statements. Thus, we have,

The negation of a conjunction p ^ q q is the disjunction of the negation of p and the negation of q. Equivalently, we write,

```~(p ^ q) ≡ ~ p v ~ q
```

## Negation of Disjunction:

A disjunction p v q consists of two sub-statements p and q which are such that either p or q or both exist. Therefore, the negation of the disjunction would mean the negation of both p and q simultaneously.

The negation of a disjunction p v q is the conjunction of the negation of p and the negation of q. Equivalently, we write,

```~(p v q) ≡ ~ p ^ ~ q
```

## Negation of Negation:

A negation of negation of a statement is the statement itself. Equivalently, we write,

```~(~p) ≡ p
```

## Negation of Implication:

If p and q are two statements, then

```~(p => q) ≡ p ^ ~ q
```

## Negation of Bicondional:

If p and q are two statements, then

```~(p ⇔ q) ≡ p ⇔ ~ q ≡ ~ p ⇔ q
```