A statement is a sentence that is either true or false, but not both. If you cannot give it a truth value, it is a non-statement.
This lesson starts the logical reasoning chapter. Everything after it, from negation to arguments, relies on this one test.
What is the test for a statement?
Ask one question: can I say this is true or say this is false? If the answer is yes, it is a statement, whatever the truth value turns out to be.
| Sentence | Statement? | Reason |
|---|---|---|
| 12 is divisible by 4 | Yes, true | checkable |
| 15 is a prime number | Yes, false | checkable, and 15 = 3 × 5 |
| Is 9 a square number? | No | a question |
| Add 6 to each side | No | a command |
| x + 3 = 10 | No | depends on x |
| Mathematics is the best subject | No | an opinion |
The two “yes” rows show that false does not disqualify a sentence. A false statement is still a statement.
Worked example: three sentences about a number
Look at these three sentences and decide what each one is.
A. “n is greater than 20.”
B. “When n = 25, n is greater than 20.”
C. “There is a number n such that n is greater than 20.”
Sentence A is an open sentence. Its truth depends on n, so it is a non-statement.
Sentence B fixes n = 25. Since 25 > 20, it is a true statement.
Sentence C adds the quantifier “there is a number”. You can find one such number, for example 21, so it is a true statement.
The example shows two ways to turn an open sentence into a statement: give the variable a value, or add a quantifier. Both ideas return in the lesson on negating statements with quantifiers.
The mistake that costs marks
The usual slip is treating any sentence with numbers in it as a statement. The expression x² − 4 = 0 has numbers and an equals sign, but it is still an open sentence until x is fixed.
| Step | Wrong | Right |
|---|---|---|
| Look at x² − 4 = 0 | has an equals sign, so a statement | ask whether it is true or false without knowing x |
| Decide | statement | non-statement |
| Fix x = 2 | not attempted | 4 − 4 = 0, so a true statement |
| Fix x = 3 | not attempted | 9 − 4 = 5, not 0, so a false statement |
The same open sentence gave one true and one false statement, which proves it cannot have a single truth value on its own.
Check yourself
Decide whether each sentence is a statement. If it is, give its truth value.
- 3 × 8 = 24
- Is 36 a perfect square?
- 2y − 1 = 7
- All triangles have four sides.
Answer
-
Statement, true, since 3 × 8 = 24.
-
Not a statement, because it is a question.
-
Not a statement, because its truth depends on y. It becomes a true statement if y = 4.
-
Statement, false, since a triangle has three sides. The quantifier “all” does not stop it from being a statement.
What to study next
Move on to negating statements with quantifiers, then test yourself with the logical reasoning practice set.
The algebra step repair trainer can help with the algebra side of open sentences. For a teacher to work through examples with you, see online one-to-one Mathematics tuition.