The negation of a statement has the opposite truth value in every situation. For statements with “all” and “some”, that means the quantifier changes as well as the verb.
This lesson follows statements and non-statements, and it sits inside the logical reasoning chapter.
How do I negate a statement with all or some?
Ask what would make the original statement false, then write that. For “all”, one exception is enough to make it false, so the negation says “some … not”.
| Original | Negation |
|---|---|
| All A are B | Some A are not B |
| Some A are B | No A are B |
| p is true | p is not true |
The second row reads naturally in plain English as “no A are B”, meaning there is no example at all.
Worked example: a statement about multiples
Consider the statement P: “All multiples of 6 are multiples of 3.”
Step 1. Decide the truth value of P. Every multiple of 6 equals 6k, which is 3 × 2k, so P is true.
Step 2. Write the negation: “Some multiples of 6 are not multiples of 3.”
Step 3. Test it with 6, 12, 18 and 24. All of them are multiples of 3, so you cannot find an exception.
The negation is false, which is the opposite of P, as it should be.
The truth-value check works because a true statement must have a false negation. If your negation were also true, you would know it was wrong.
The mistake that costs marks
The common slip is swapping “all” with “none” instead of “some … not”. It feels like the opposite, but it is too strong.
Take Q: “All the students in a group of five have a bicycle.” Suppose three of the five have a bicycle.
| Step | Wrong | Right |
|---|---|---|
| Negation written | None of the students has a bicycle | Some students do not have a bicycle |
| Q in this group | false (only three have one) | false (only three have one) |
| Negation in this group | false, since three have one | true, since two do not |
The wrong negation and the original are both false in this group. A correct negation must be true when the original is false, so the wrong one is ruled out.
Check yourself
Write the negation of each statement and state which one is true.
- All squares have four equal sides.
- Some even numbers are prime.
- 5 is greater than 9.
Answer
-
Negation: “Some squares do not have four equal sides.” The original is true, so the negation is false.
-
Negation: “No even numbers are prime.” The original is true, because 2 is even and prime, so the negation is false.
-
Negation: “5 is not greater than 9.” The original is false, so the negation is true.
What to study next
Next, write implications and their converse, which uses the same truth-value check. Then try the logical reasoning practice set.
A mistake log is a good place to record negations you wrote wrongly. For lessons with a teacher, see online one-to-one Mathematics tuition.