Prove ${\forall x \; \forall y \; (x + y = y + x)}$

Question

Question:

Determine the truth value of the statement if the universe of each variable consists of all the integers.

Give reason to your answer if the statement is true and provide a counterexample for the false statement.

${\forall ‌x \; \forall ‌y \; (x + y = y + x)}$ $$\tag*{$(2\;marks)$}$$

Answer:

True.

${Suppose\;x = m,\;y = n,\;m,\;n \in Z}$

${By\;defination\;of\;commutativity,\; m + n = n + m}$

${Then\;x + y = y + x}$

${\;\;\;\;\;\;\;\;\;m + n = n + m}$

Can I prove like this?

Answer 1

maybe you can use the successor function for the integers

where " s " is the successor function

x = s(x-1) and y = s(y-1)

x + y = s(x-1) + y = s(x + y -1)

y + x = s(y-1) + x = s(y + x -1)

Now it remains to know if "s" is a function that is commutative

I think you can not prove it so because this is a particular case where x = y

Answer 2

If you are given commutivity, the result follows immediately.

If you are given the Peano axioms and the definition of addition in terms of the successor function, you should look at Landau's "Foundations of Analysis" (do a Google search).