Show that a class of nine students must have at least three male students or at least seven female students.

Question

I am stuck with the following problem: Suppose that there are nine students in a class. Show that the class must have at least three male students or at least seven female students.

Please help me to solve this. Thanks.

Answer 1

Proof by contradiction:

Define $p$ to be the statement that the class has at least three male students, and $q$ that the class has at least seven female students. So we need to prove that $p \lor q$ is true.

If $p \lor q$ were not true, that is, $\lnot (p \lor q)$ is true. Note that

$$\lnot (p \lor q) \Leftrightarrow (\lnot p) \land (\lnot q),$$

where $\lnot p$ means that the class has less than three male students and $\lnot q$ that the class has less than seven female students. Now the class has at most eight students, which leads to contradiction.