Mathematical statements and truth

• A mathematical statement is a sentence that has a definite truth value: it is either true or false. "2 + 2 = 4" is a statement, and "x + 2 = 4" is not a statement by itself because it depends on what x is. Math cares a lot about being precise about what is being claimed, so we separate statements from expressions, questions, and commands. Many statements come with conditions like "if..." or with hidden assumptions like "let x be a real number." When you read a sentence in math, your first job is to ask: does it claim something that can be checked as true or false given the rules of the system?
• A key idea is the difference between proving and disproving. A universal claim like "all integers have property P" is disproved by finding one counterexample, a single integer that fails the property. But proving a universal claim is harder: you must give a reasoning chain that works for every allowed input, not just examples. This is why proofs rely on definitions and algebraic steps rather than checking cases randomly. Beginners often mistake "I tested it a few times" for proof. Testing builds intuition, but a proof is what guarantees the claim for all cases.

• "2 + 2 = 4" is a statement (true).
• "x + 2 = 4" is not a statement (depends on x).
• "For all real x, x + 2 > x" is a true statement.

Classify each as statement or not. If statement, say true or false.

• 19 is prime
• 2What time is it
• 3For all integers n, n^2 >= n
• 4x^2 = 9
• 5There exists an integer n with n^2 = 2
• 6If it rains, the ground is wet
• 7Every even integer is divisible by 4
• 80 < 1

MCQ

Which is a statement?