Consider the following proof exercise:
Show that x+0=0 for every xϵZ
Proof:
Let x=[a,b] and recall that 0=[1,1].
Recall the definition of integer addition i.e. [r,s]+[t,u]=[r+t,s+u].
So, x+0=[a,b]+[1,1]=[a+1,b+1].
We need to show that [a+1,b+1]=[a,b].
Recall that Z is the set of equivalence classes of R where R is the relation on X such that (a,b)R(c,d) means a+d=b+c and X is the set of ordered pairs (a,b) of natural numbers.
(a+1,b+1)R(a,b) since a+1+b=b+1+a
Thus, [a+1,b+1]=[a,b] and the result holds.
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