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Sentence (Mathematical) — Definition, Formula & Examples

A mathematical sentence is a complete statement involving numbers, variables, or expressions connected by a relation symbol (like ==, <<, or >>) that can be judged as true or false.

A mathematical sentence is a well-formed expression in mathematics that asserts a relationship between quantities using a verb (such as equals, is less than, or is greater than) and that possesses a definite truth value — it is either true or false, but not both.

How It Works

To identify a mathematical sentence, check two things: it must contain a relation symbol (a mathematical "verb" like ==, ≠\neq, <<, >>, ≤\leq, or ≥\geq), and it must make a claim you can evaluate as true or false. For example, 3+4=73 + 4 = 7 is a mathematical sentence because it states a claim that is true. By contrast, 3+43 + 4 is a mathematical expression — it names a value but makes no claim, so it is not a sentence. A sentence with a variable, like x+2=9x + 2 = 9, is called an open sentence because its truth depends on the value of xx.

Example

Problem: Classify each of the following as a mathematical sentence or not, and if it is a sentence, state whether it is true or false: (a) 5+3=85 + 3 = 8, (b) 6×26 \times 2, (c) 10>1510 > 15.
Item (a): 5+3=85 + 3 = 8 contains the relation symbol == and makes a claim. Since 5+35 + 3 does equal 88, this is a true mathematical sentence.
5+3=8(true sentence)5 + 3 = 8 \quad \text{(true sentence)}
Item (b): 6×26 \times 2 is an expression. It computes to 1212, but it does not make any claim — there is no relation symbol. It is not a mathematical sentence.
Item (c): 10>1510 > 15 contains the relation symbol >> and claims that 1010 is greater than 1515. That claim is false, but it is still a mathematical sentence because it has a definite truth value.
10>15(false sentence)10 > 15 \quad \text{(false sentence)}
Answer: (a) true sentence, (b) not a sentence (it is an expression), (c) false sentence.

Why It Matters

Understanding the difference between sentences and expressions is essential in algebra, where you solve open sentences (equations and inequalities) by finding values that make them true. In logic and proof courses, recognizing which statements carry truth values is the foundation for constructing valid arguments.

Common Mistakes

Mistake: Confusing a mathematical expression with a mathematical sentence.
Correction: An expression like 2x+52x + 5 names a quantity but makes no claim. A sentence requires a relation symbol (e.g., 2x+5=112x + 5 = 11) so it can be true or false.