Translating

Example 2.8

Theorem: If \(f:\mathbb{R}\to\mathbb{R}\) is continuous on the interval \([a,b]\) and differentiable on \((a,b)\), then there is a number \(c\in(a,b)\) for which \[ f'(c) = \frac{f(b)-f(a)}{b-a} \]

Example 2.9

Conjecture: Every even integer greater than \(2\) is the sum of two primes.

Problem 2.3

If \(x\) is prime then \(\sqrt{x}\) is not rational.

Textbook answer: \(P\implies \sim Q\) where \(P(x)\) is “\(x\) is prime” and \(Q(x)\) is “\(\sqrt{x}\) is a rational number.”

Alternative:

Problem 2.13

Everything is funny as long as it is happening to someone else.

Textbook answer : \[ \forall x, (\sim M(x)\wedge S(x))\implies F(x) \] where \(M(x)\) means “\(x\) is happening to me”, \(S(x)\) is “\(x\) is happening to someone”, \(F(x)\) means “\(x\) is funny.”