Which of the following is not true?

A set S is closed under an operation if applying that operation to any two elements of S always produces a result that still belongs to S. For addition, this means: for all a, b in S, a + b is also in S. To judge whether a closure statement is true, either prove it holds for every pair, or find a single counterexample where it fails.

1. Integers: the integers (…, −2, −1, 0, 1, 2, …) are closed under addition. For example, 3 + 5 = 8 and −4 + 9 = 5, both integers. So "the sum of two integers is an integer" is true.

2. Rational numbers: for rationals a = p/q and b = r/s (integers p, q, r, s with q, s ≠ 0), a + b = (ps + rq)/(qs), which is again a ratio of integers with a nonzero denominator, so it is rational. So "the sum of two rational numbers is a rational number" is true.

3. Irrational numbers: this closure fails. √2 and −√2 are each irrational, but √2 + (−√2) = 0, which is rational, not irrational. So "the sum of two irrational numbers is an irrational number" is not true.

4. Natural numbers: the natural numbers (1, 2, 3, …) are closed under addition. For example, 4 + 7 = 11, a natural number. So "the sum of two natural numbers is a natural number" is true.

A second example confirms the irrational case is not a one-off coincidence: 3 + √5 and 3 − √5 are each irrational, yet their sum is (3 + √5) + (3 − √5) = 6, a rational number. Irrational parts can cancel, so the set of irrational numbers is not closed under addition.

Three of the four statements always hold, and only the statement about irrational numbers fails in general. So the statement that is not true is: "The sum of two irrational numbers is an irrational number."