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.