Equivalence relations are meant to answer the question, “When are two elements of this set the same relative to some special property?” The most common equivalence relation is numerical equality: two numbers are equal if they have the same value.
Example B.2.2.
Given integers \(a,b,c,d\) where \(b,d\) are nonzero, we say \((a,b) \sim (c,d)\) provided \(ad - bc = 0\text{.}\)
To verify that \(\sim\) is an equivalence relation, we’ll check the three properties. To start, suppose \(a,b,c,d,e,f\) are integers where \(b,d,f\) are nonzero. For reflexivity, observe \((a,b) \sim (a,b)\) because \(ab - ab = 0\text{.}\)
For symmetry, observe that \((a,b) \sim (c,d)\) implies \((c,d) \sim (a,b)\) because \(ca - db = ac - bd = 0\) by the commutativity of integer multiplication our starting assumption.
For transitivity, observe that \((a,b) \sim (c,d)\) and \((c,d) \sim (e,f)\) implies \((a,b) \sim (e,f)\) because \(ad - bc = 0\) and \(cf - de = 0\) imply \(0 = adf - bde = (af-be)d \text{,}\) which implies \(af-be = 0\) (recall, \(d \not = 0\) by assumption).
This equivalence relation is important! When we talk about the rational numbers, we are really talking about equivalence classes of ordered pairs of integers modulo \(\sim\text{,}\) where the first element of the ordered pair is the what we usually call the numerator of the rational number, and the second (nonzero) element of the ordered pair is what we usually call the denominator.
