9. В классе учится 25 человек, из них 18 человек посещают математический кружок, а 12 — кружок по астрономии. Выберите верные утверждения и запишите в ответе их номера.

Let M be the set of students attending the math club and A be the set of students attending the astronomy club. |Total| = 25, |M| = 18, |A| = 12.

Using the principle of inclusion-exclusion: |M U A| = |M| + |A| - |M ∩ A|. Since |M U A| <= 25, we have 18 + 12 - |M ∩ A| <= 25, which means 30 - |M ∩ A| <= 25, so |M ∩ A| >= 5.

1) False, because if everyone attended both, |M ∩ A| = 25, but |M| + |A| = 30, so at least 5 must attend both.

2) True, since |M ∩ A| >= 5, there are at least 5 students attending both.

3) False, as shown above, not everyone in M is necessarily in A.

4) True, since |M ∩ A| >= 5, and 5 < 13.

Answer: 2, 4