What does it mean to say that a number \(x\) is *irrational*?

Prove by contradiction statements A and B below, where \(p\) and \(q\) are real numbers.

**A:** If \(pq\) is irrational, then at least one of \(p\) and \(q\) is irrational.

**B:** If \(p+q\) is irrational, then at least one of \(p\) and \(q\) is irrational.

Disprove by means of a counterexample statement C below, where \(p\) and \(q\) are real numbers.

**C:** If \(p\) and \(q\) are irrational, then \(p+q\) is irrational.

If the numbers \(e\), \(\pi\), \(\pi^2\), \(e^2\) and \(e\pi\) are irrational, prove that at most one of the numbers \(\pi+e\), \(\pi-e\), \(\pi^2-e^2\), \(\pi^2+e^2\) is rational.