# Negation Statement

How to find the negation of a statement and its truth value: definition, truth value, 6 examples, and their solutions.

## Definition

### Statement

A statement is a sentence

that has one truth value:

true or false.

We use [p] to symbol a statement.

These sentences all either true or false.

So these sentences are all statements.

### Negation

[~p] is the negation of a statement.

It means [not p].

To find ~p,

negate the statement p.

## Example 1

### Example

### Solution

The given statement is

5 is a positive number.

So the negation is

5 is not a positive number.

So

5 is not a positive number

is the answer.

## Example 2

### Example

### Solution

The given statement is

1 + 2 = 0.

So the negation is

1 + 2 ≠ 0.

So

1 + 2 ≠ 0

is the answer.

## Example 3

### Example

### Solution

The given statement is

2 is not an odd number.

The statement is already negated:

[not] is already in the statement.

Then, to negate the statement,

remove the [not]:

2 is an odd number.

So

2 is an odd number

is the answer.

## Truth Value

### Truth Table

This is a truth table

that shows the truth values

of a statement [p] and its negation [~p].

As you can see,

p and ~p have the opposite truth values.

If p is true (T),

then ~p is false (F).

If p is false (F),

then ~p is true (T).

## Example 4

### Example

### Solution

p: 5 is a positive number.

This is true.

p and ~p have the opposite truth values.

p is true.

So ~p is false.

So false is the answer.

## Example 5

### Example

### Solution

q: 3 is an even number.

This is false.

q and ~q have the opposite truth values.

q is false.

So ~q is true.

So true is the answer.

## Example 6

### Example

### Solution

r: 2 is a prime number.

This is true.

~(~r) means double negation.

A statement is negated twice.

So ~(~r) comes back to the original r:

~(~r) = r.

It's like (-)⋅(-) = (+).

Multiply Negative Numbers

r is true.

So ~(~r) is true.