Questions: Truth Values and Statements

Question 1

Which of the following is a mathematical statement?

• A. Solve for x in the equation x + 5 = 12
• B. Is 17 a prime number?
• C. The square root of two is rational
• D. x + 3 = 7

Answer: (C)

A statement is a declarative sentence that is either true or false. 'The square root of two is rational' is a statement — it is false. Option A is a command (no truth value). Option B is a question (no truth value). Option D is an open sentence (predicate) — it depends on x and has no truth value until x is specified or quantified. A common misconception is that option D is a statement because it 'looks mathematical,' but it is a predicate: it becomes true for x=4 and false for x=5.

Question 2

A student argues that 'x > 0' becomes a statement when we substitute x = 3. The student is:

• A. Wrong — open sentences can never become statements, even with substitution
• B. Correct — substituting x = 3 gives '3 > 0', which is a true statement
• C. Wrong — we must also know the domain before it has a truth value
• D. Correct — any equation involving numbers automatically qualifies as a statement

Answer: (B)

Substituting a specific value for the variable converts a predicate into a statement. '3 > 0' is a genuine statement — it is true. The key distinction is between a predicate like 'x > 0' (truth depends on x, so no single truth value) and a statement like '3 > 0' (definite truth value regardless of context). Option A is wrong: substitution is precisely one of the two ways to produce a statement from a predicate. Option C is wrong for this example — once x is replaced by a specific number, no domain ambiguity remains.

Question 3

A false sentence is not a mathematical statement — it is expected to be true to count as a statement.

• T. True
• F. False

Answer: False

This is one of the most common misconceptions about statements. A statement is defined by its *capacity* to have a truth value, not by the value itself. '5 > 10' is a perfectly good mathematical statement — it is simply false. '2 + 2 = 5' is a statement; it is false. The truth value of a statement can be true or false; what disqualifies a sentence from being a statement is being a question, command, or open sentence — not being false.

Question 4

The sentence 'For all integers x, x + 3 = 7' is a statement.

• T. True
• F. False

Answer: True

Binding the variable x with the universal quantifier 'for all integers x' converts the predicate 'x + 3 = 7' into a statement. The statement has a definite truth value: it is false (it fails for x = 1, 2, 3, and almost every other integer). The key is that the quantifier removes the dependence on a free variable — nothing is unspecified, so the sentence is either true or false. This is one of the two ways (the other being substitution) to turn a predicate into a statement.

Question 5

Why is 'x + 3 = 7' not a statement on its own, but 'for all integers x, x + 3 = 7' is a statement? What changes?

Model answer: The open sentence 'x + 3 = 7' contains a free variable x — its truth depends on what x is, so it has no single truth value and is therefore a predicate, not a statement. The universal quantifier 'for all integers x' binds the variable, removing the dependence on any particular value. The result has a definite truth value (false — since it fails for x = 1, among others), making it a genuine statement.

The distinction between predicates and statements is foundational to all of logic. Proofs consist of statements — sentences with definite truth values. A predicate like 'x + 3 = 7' is a function from values to truth values, not a truth value itself. Quantifiers (for all, there exists) convert predicates into statements by either ranging over all possible values or asserting the existence of one that satisfies the condition. Understanding this is the prerequisite for working with quantified logic, the main language of mathematical proof.