mathematical statement
Noun: A mathematical statement is a declarative sentence or expression that asserts a mathematical fact, relation, or property. It is a precise formulation within the language of mathematics that can be objectively determined to be true or false.
A mathematical statement is a formal assertion. It is the fundamental unit used to express theorems, propositions, lemmas, and corollaries. Unlike a general opinion or command, its truth value is not subjective.
- The equation "2 + 2 = 4" is a true mathematical statement.
- "The square root of 2 is a rational number" is a false mathematical statement.
- "For all real numbers , ² ≥ 0" is a mathematical statement known as a theorem.
- "Let be an integer" is not a mathematical statement by itself, as it is a definition or hypothesis, not a declarative assertion with a truth value.
- In logic, a mathematical statement is often called a "proposition." The field of proof theory analyzes the structure and derivability of mathematical statements.
- A mathematical statement can be conditional (e.g., "If a triangle is equilateral, then it is equiangular") or universal (e.g., "Every even integer greater than 2 can be expressed as the sum of two primes").
- Theorem: A mathematical statement that has been proven to be true.
- Proposition: A mathematical statement that is less significant than a theorem but is still important and proven.
- Lemma: A mathematical statement proven primarily to help in proving a larger theorem.
- Corollary: A mathematical statement that follows readily from a previously proven theorem.
- Conjecture: A mathematical statement that is proposed to be true but has not yet been proven.
- Proposition
- Assertion (in a mathematical context)
- Formula (when it expresses a relation, e.g., E=mc²)
Not every sentence in a mathematical text is a mathematical statement. Commands ("Solve for x"), questions ("Is this prime?"), and definitions ("Let epsilon be greater than zero") are not statements because they cannot be assigned a truth value of true or false. A mathematical statement must be a declarative claim.
-
a statement of a mathematical relation
