The language of additive monoids has a constant symbol 0 and

The language of additive monoids has a constant symbol 0 and a binary function symbol, written x + y. The axioms are associativity Forall x Forall y Forall z (x + (y + z) = (x + y) + z); commutativity Forall x Forall y(x + y = y + x); and 0 is a (two-sided) unit Forall x (0 + x = x) and Forall x (x + 0 = x). Define the even elements by E(x) = _df z(x = z + z). Use deduction to show that the sum of two even elements is even.

Solution

let we have x such that there exist some natural number z for which x = z + z = 2z.

and y such that there exist some natural number w for which y = w + w = 2w.

=> x and y both are even.

Let we have a such that a = x + y = 2z + 2w = 2(z + w).

Since a is a multiple of 2 then we can say a is even.

=> sum of 2 even numbers is even.