Page 230 - HOW TO PROVE IT: A Structured Approach, Second Edition

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216 Relations
S is reflexive. To see that S is symmetric, suppose (x, y) ∈ S. By the definition
of S, this means that x − y ∈ Z. But then y − x =−(x − y) ∈ Z too, since
the negative of any integer is also an integer, so (y, x) ∈ S. Because (x, y)
was an arbitrary element of S, this shows that S is symmetric. Finally, to see
that S is transitive, suppose that (x, y) ∈ S and (y, z) ∈ S. Then x − y ∈ Z
and y − z ∈ Z. Because the sum of any two integers is an integer, it follows
that x − z = (x − y) + (y − z) ∈ Z,so(x, z) ∈ S, as required. Thus, S is an
equivalence relation on R.
What do the equivalence classes for this equivalence relation look like? We
have already observed that (5.73, 2.73) ∈ S and (−1.27, 2.73) ∈ S,so5.73 ∈
[2.73] and −1.27 ∈ [2.73]. In fact, it is not hard to see what the other elements
of this equivalence class will be:
[2.73] ={..., −1.27, −0.27, 0.73, 1.73, 2.73, 3.73, 4.73, 5.73,...}.

In other words, the equivalence class contains all positive real numbers of the
form “ .73” and all negative real numbers of the form “– .27.” In general,
for any real number x, the equivalence class of x will contain all real numbers
that differ from x by an integer amount:
[x] ={..., x − 3, x − 2, x − 1, x, x + 1, x + 2, x + 3,...}.

Here are a few facts about these equivalence classes that you might try to
prove to yourself. As you can see in the last equation, x is always an element
of [x]. If we choose any number x ∈ [2.73], then [x] will be exactly the same
as [2.73]. For example, taking x = 4.73 we find that
[4.73] ={..., −1.27, −0.27, 0.73, 1.73, 2.73, 3.73,
4.73, 5.73,...}= [2.73].

Thus, [4.73] and [2.73] are just two different names for the same set. But if we
choose x ∈ [2.73], then [x] will be different from [2.73]. For example,

[1.3] ={..., −1.7, −0.7, 0.3, 1.3, 2.3, 3.3, 4.3,...}.

In fact, you can see from these equations that [1.3] and [2.73] have no elements
in common. In other words, [1.3] is actually disjoint from [2.73]. In general,
for any two real numbers x and y, the equivalence classes [x] and [y] are either
identical or disjoint. Each equivalence class has many different names, but
different equivalence classes are disjoint. Because [x] always contains x as an
element, every equivalence class is nonempty, and every real number x is in
exactly one equivalence class, namely [x]. In other words, the set of all of the
equivalence classes, R/S, is a partition of R. This is another illustration of the