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The Integers Modulo n 189
11.4 The Integers Modulo n
Example 11.8 proved that for a given n ∈ N, the relation ≡ (mod n) is
reflexive, symmetric and transitive, so it is an equivalence relation. This
is a particularly significant equivalence relation in mathematics, and in
the present section we deduce some of its properties.
To make matters simpler, let’s pick a concrete n, say n = 5. Let’s begin
by looking at the equivalence classes of the relation ≡ (mod 5). There are
five equivalence classes, as follows:
[0] = { x ∈ Z : x ≡ 0 (mod 5)) } = { x ∈ Z : 5 | (x − 0) } = { ...,−10,−5,0,5,10,15,... } ,
[1] = { x ∈ Z : x ≡ 1 (mod 5)) } = { x ∈ Z : 5 | (x − 1) } = { ..., −9,−4,1,6,11,16,... } ,
[2] = { x ∈ Z : x ≡ 2 (mod 5)) } = { x ∈ Z : 5 | (x − 2) } = { ..., −8,−3,2,7,12,17,... } ,
[3] = { x ∈ Z : x ≡ 3 (mod 5)) } = { x ∈ Z : 5 | (x − 3) } = { ..., −7,−2,3,8,13,18,... } ,
[4] = { x ∈ Z : x ≡ 4 (mod 5)) } = { x ∈ Z : 5 | (x − 4) } = { ..., −6,−1,4,9,14,19,... } .
Notice how these equivalence classes form a partition of the set Z.
We label the five equivalence classes as [0],[1],[2],[3] and [4], but you
know of course that there are other ways to label them. For example,
[0] = [5] = [10] = [15], and so on; and [1] = [6] = [−4], etc. Still, for this
discussion we denote the five classes as [0],[1],[2],[3] and [4].
These five classes form a set, which we shall denote as Z 5 . Thus
Z 5 = { [0],[1],[2],[3],[4] }
is a set of five sets. The interesting thing about Z 5 is that even though its
elements are sets (and not numbers), it is possible to add and multiply
them. In fact, we can define the following rules that tell how elements of
Z 5 can be added and multiplied.
[a] + [b] = [a + b]
[a] · [b] = [ a · b ]
For example, [2] + [1] = [2 + 1] = [3], and [2] · [2] = [2 · 2] = [4]. We stress
that in doing this we are adding and multiplying sets (more precisely
equivalence classes), not numbers. We added (or multiplied) two elements
of Z 5 and obtained another element of Z 5 .
Here is a trickier example. Observe that [2] + [3] = [5]. This time we
added elements [2],[3] ∈ Z 5 , and got the element [5] ∈ Z 5 . That was easy,
except where is our answer [5] in the set Z 5 = { [0],[1],[2],[3],[4] } ? Since
[5] = [0], it is more appropriate to write [2] + [3] = [0].