Exercise 8.24

Arithmetic assertions can be written in first-order logic with the predicate symbol $<$, the function symbols ${+}$ and ${\times}$, and the constant symbols 0 and 1. Additional predicates can also be defined with biconditionals.

1. Represent the property “$x$ is an even number.”

2. Represent the property “$x$ is prime.”

3. Goldbach’s conjecture is the conjecture (unproven as yet) that every even number is equal to the sum of two primes. Represent this conjecture as a logical sentence.