Negations

The negation of a statement is the statement formed by putting the words “it is not true that” in front of the original statement. In logic the symbol ~ is used for negation. (Again, you won’t see that symbol on the Math 1 test.)

Key Fact R1

A statement and its negation have opposite truth values:

• If p is true, ~p is false.
• If p is false ~p is true.

Here are the negations of statements p, q, and r given above:

• ~p: It is not true that 2 + 2 = 4
         or equivalently: 2 + 2 ≠ 4.
• ~q: It is not true that every prime number is odd
         or equivalently: There is a prime number that is not odd.
• ~r: It is not true that there is a real number x such that x² + 1 = 0
         or equivalently: There is no real number x for which x² + 1 = 0.

Note that ~p is false and ~q and ~r are true.

Key Fact R2

If a statement claims that all objects of a certain type have a particular property, the negation of that statement says that at least one of those objects does not have the property.

For example, statement q (falsely) asserts that all prime numbers are odd. Its negation, ~q, (truthfully) states that there is at least one prime number that is not odd (2 is prime, but not odd).

Similarly, the negation of the statement “All roses are red” is the statement “There is at least one rose that is not red.”

Key Fact R3

If a statement claims that some object of a certain type has a particular property, the negation of that statement says that none of those objects has the property.

For example, statement r (falsely) asserts that some real number is a solution of the equation x² + 1 = 0. Its negation, ~r, (truthfully) states that no real number is a solution of the equation x² + 1 = 0.

Similarly, the negation of the statement “Some roses are black” is the statement “No roses are black.”