Negating statement is mathematics is sometimes tricky. The following is a list of typical situations where you need to be careful when negating logical statements. If you can analyze the following typical sentences or statements, you’ll be able to use “the way the statement has been negated” in similar situations:

All cars are red. Negation: Not all cars are red. Or, There is at least one car which is not red.

All cars are not red. Negation: All cars are red.

Some prime numbers are odd. Negation: All prime numbers are not odd.

No student is lazy. Negation: Some students are lazy.

Some cats are not black. Negation: All cats are black.

Every natural number is an integer. Negation: Every natural number is not an integer.