Negation Statement
How to find the negation of a statement and its truth value: definition, truth value, 6 examples, and their solutions.
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
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
The given statement is
1 + 2 = 0.
So the negation is
1 + 2 ≠ 0.
So
1 + 2 ≠ 0
is the answer.
Example
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.
Thuth Value
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
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
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.
ExampleDouble Negation
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.