Homework 8 Due Tuesday, March 14. Define the predicates Even(x) and Odd(x) in the language of our theory of arithmetic (this is easy). Prove the theorem (Ax.Even(x) <+> Odd(x)) formally using only axioms and perhaps theorems proved in class. It is permissible to prove this as two theorems: (Ax.(Even(x) | Odd(x))) and (Ax.(~(Even(x) & Odd(x)))) if you don't like working with xor! Comments added after March 15 discussion: proving either (Ax.(Even(x) | Odd(x))) or (Ax.(~(Even(x) & Odd(x)))) by itself is acceptable. The second one is harder -- it needs the theorem that each nonzero natural number has a predecessor.