 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/discrete-mathematics/3/c/iii/solution

Every nonzero integer

a

has

a^{2} > 0

, proving reflexivity. Because

ab = ba

, the relation is symmetric. If

ab > 0

and

b c > 0

, then

a c = (ab) (b c) / b^{2} > 0

, proving transitivity. Thus this is an equivalence relation, whose two equivalence classes are the positive integers and the negative integers:

{1, 2, 3, \dots}, {- 1, - 2, - 3, \dots} .

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