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