Negation of a Statement

For some simple statements we can construct the negation just by adding the word "not".

Statement Negation of the Statement
x is equal to 3 ( x = 3 )
 x is not equal to 3 ( x ≠ 3 )
x is less than 5 ( x < 5 ) x is not less than 5 ( x
x is greater than or equal to 5 ( x ≥ 5 )

When the statement contains words such as "all", "no", or "some," then its negation is more complicated.

Statement Negation of the Statement
All x satisfy A.
Every x satisfies A.
 At least one x does not satisfy A.
There is an x which does not satisfy A.
Some x does not satisfy A.
No x satisfies A.
Every x does not satisfy A.
At least one x satisfies A.
Some x satisfies A.
There is an x which satisfies A.
At least one x satisfies A.
Some x satisfies A.
 No x satisfies A.
Every x does not satisfy A.

We can also negate compound statements containing "and" and "or".

Statement Negation of the Statement
A and B are both true.
 At least one of A or B is not true.
A and B and C are all true.
 At least one of A or B or C is not true.
A or B is true.
 Both A and B are not true.

Practice 2: Write the negation of each of these statements.
(a) x + 5 ≥ 3
(b) All prime numbers are odd.
(c) x^2 < 4
(d) x divides 2 and x divides 3.
(e) No mathematician can sing well.