A relationR on a set A is said to be an equivalence relation if it is reflexive, symmetric, and transitive.
For example, each of the following is an equivalence relation:
• = (equality) on any set A;
• R defined on Z by mRn iff m − n is even;
• R defined on N by (m, n)R(p, q) iff mq = np.
A B C
Figure 5.3: Graph of an equivalence relation.
The proof that each of these is an equivalence relation is left to the reader. In each case there are three conditions to verify, of course.
In the case of an equivalence relation, the associated digraph exhibits a very special form, illustrated by Figure 5.3. Namely, the graph divides up into distinct subgraphs, each of which is a reflexive, symmetric, transitive graph in which every pair of ver-tices is joined by an edge, and no two of which subgraphs are joined by any edge. In the example illustrated, these subgraphs are labeled A, B, C.
If R is an equivalence relation on a set A, then for any a ∈ A we define the equivalence class ofa (modulo R) to be the set:
[a] = {x ∈ A | aRx}
For example, let R be the equivalence relation defined on the setN of natural numbers bymRn iff m− n is even. Then, for instance,
[1] = {n | mRn} = {n | 1−n is even} = {n | n is odd}
and
[2] = {n | mRn} = {n | 2−n is even} = {n | n is even}
Continuing in this manner, we see that:
[1] = [3] = [5] = [7] = . . . = the odd numbers [2] = [4] = [6] = [8] = . . . = the even numbers
Thus in this case there are just two equivalance classes, the set of even numbers and the set of odd numbers. The equivalence class of any odd number is the set of all odd numbers, and the equivalence class of any even number is the set of all even numbers. In particular, this particular equivalence relation splits the underlying set N into two disjoint parts.
The following theorem shows that this is just a special case of a quite general phenomenon.
THEOREM 5.5.1
LetR be an equivalence relation on a set A.
(i) For anya∈ A, a ∈ [a].
(ii) For anya, b∈ A, aRb iff [a] = [b].
(iii) For anya, b∈ A, if ¬(aRb) then [a] ∩ [b] = ∅.
(iv) Any two equivalence classes are either equal or disjoint.
PROOF
(i) SinceR is reflexive, aRa, so
a∈ {x ∈ A | aRx} = [a]
(ii) Assume[a] = [b]. By (i), b ∈ [b]. Thus b ∈ [a]. That is, b∈ {x ∈ A | aRx}
ThusaRb.
Conversely, assumeaRb. We prove that[a] = [b]. Let x ∈ [a] be arbitrary.
Then by definition,aRx. Now by symmetry, our assumption gives bRa. Since bRa and aRx, transitivity gives bRx. Thus x ∈ [b]. Now suppose x ∈ [b].
Then by definition,bRx. Thus we have aRb and bRx, so by transitivity, aRx, givingx∈ [a]. The proof of (ii) is complete.
(iii) We prove the (logically equivalent) contrapositive, namely that if[a]∩[b] = ∅ thenaRb. Let x∈ [a] ∩ [b]. Since x ∈ [a], aRx. Since x ∈ [b], bRx, so by symmetry,xRb. Then aRx and xRb, so by transitivity, aRb.
(iv) For any paira, b∈ A, either aRb or ¬(aRb). If aRb then by (ii), [a] = [b].
If¬(aRb), then by (iii), [a] ∩ [b] = ∅.
By a partition of a setA we mean a collectionF of pairwise disjoint subsets of A (i.e.,X∩ Y = ∅ for every distinct pair of sets X, Y in F) whose union is the whole ofA, i.e.,
A=
F
For example, ifP is the set of all prime numbers, Q is the set of all composite numbers, andR= {1}, then the collection {P, Q, R} is a partition of N .
Theorem 5.5.1 tells us that for any equivalence relation on a setA, the collection of all equivalence classes constitutes a partition ofA. By part (i) of the theorem, A is equal to the union of all the equivalence classes. And by part (iv), the equivalence classes are pairwise disjoint.
This result has a converse, namely:
THEOREM 5.5.2
LetF be a partition of a set A. Define a relation R on A by aRb iff (∃X ∈ F)[a, b ∈ X]
ThenR is an equivalence relation on A. Moreover, the equivalence classes for R are precisely the sets in the partitionF.
PROOF Left as an exercise.
The above observations tell us that equivalence relations are just a particular way of looking at an extremely familiar phenomenon: that of classification or categori-zation.
In many walks of life, as well as in mathematics, we frequently classify objects according to some criterion of equivalence. For instance, given a ranking of football teams or chess players, we often split them into leagues or divisions according to ability and performance. The equivalence relation involved here is one of “equiva-lent (or comparable) ability” and the equivalence classes are the different leagues or divisions.
Again, a number of U.S. publishers produce annual listings of American colleges ranked according to their degree of entrance selectivity. The equivalence relation here is one of “having roughly the same overall entrance requirements”; the equivalence classes are the groupings of colleges into categories (generally) known as “most competitive”, “highly competitive”, “very competitive”, “competitive”, “less com-petitive”, and “non-competitive”. A potential student who is told that two particular colleges are equally difficult to get into (aRb) will know that they are in the same category ([a] = [b]); conversely, knowing that two colleges are both, say, ranked
“highly competitive”, (a, b in the same partition class) a potential student will know that they are equally hard to get into (aRb).
The exercises below give some more mathematical examples of equivalence class-es, and should provide some indication of the importance of this concept in modern mathematics.
Exercises 5.5
(1) DefineR onN by mRn iff 3 divides m − n.
(a) Prove thatR is an equivalence relation onN . (b) What are [1], [2], [3]?
(c) What are [4], [5], [6]?
(d) How many equivalence classes are there altogether? Prove your answer.
(2) DefineR onN by mRn iff 3 divides m + n. Prove that R is not an equivalence relation onN .
(3) DefineR onR by: xRy iff x2= y2.
(a) Prove thatR is an equivalence relation onR.
(b) What are [0], [2], [49]?
(c) Prove that there are infinitely many equivalence classes.
(4) Fix an integerp≥ 1, and define R on Z by: mRn iff p divides m − n.
(a) Prove thatR is an equivalence relation onZ.
(b) What are [0], [1], [−1], [p], [−p], [p + 1]?
(c) Prove that [0], [1],. . . , [p− 1] are all distinct equivalence classes.
(d) Prove that [0], [1],. . . , [p− 1] are all the equivalence classes.
(5) DefineR onN × N by:
(m, n)R(p, q) iff mq= np (a) Prove thatR is an equivalence relation onN × N . (b) Give three members of each of the equivalence classes
[(1, 1)], [(1, 2)], [(2, 5)]
(c) Give a complete description of the set[(p, q)] for any pair p, q from N . (6) LetF be the collection of all finite sets. Define ≈ on F by:
X≈ Y iff there is a bijection f : X → Y (a) Prove that≈ is an equivalence relation on F.
(b) What are the equivalence classes of the sets∅, {∅}, {1, 2, 3}?
(c) What is the equivalence class of the set{a1, . . . , an} for any distinct ob-jectsa1, . . . , an?
(d) Show that there is a bijection fromN onto the collection, C, of all equiv-alence classes.
(e) Can you suggest a possible formal definition of the integers 0, 1, 2, . . . ? The relation≈ is known as equipollence. For a generalization to infinite sets, see Sections 4.6 and 4.7.
(7) DefineR onR × R by: (x, y)R(u, v) iff 3x − y = 3u − v.
(a) Prove thatR is an equivalence relation onR × R.
(b) What are[(4, 5)], [(0, 0)]?
(c) Describe[(a, b)] for any fixed values of a, b, both in set-theoretic terms and geometrically.
(8) DefineR onR × R by (x, y)R(u, v) iff x2+ y2= u2+ v2. (a) Prove thatR is an equivalence relation onR × R.
(b) What are[(0, 0)], [(1, 1)], [(3, 4)]?
(c) Describe[(a, b)] for any fixed values of a, b, both in set-theoretic terms and geometrically.
(9) Prove Theorem 5.5.2.
(10) Describe the equivalence relation onZ determined by the partition {A, B}, whereA= {x | x ≤ 0}, B = {x | x > 0}.
(11) Describe, as sets of ordered pairs, the equivalence relations on the setA = {1, 2, 3, 4, 5} associated with the following partitions:
(a) {{1, 2}, {3, 4, 5}}
(b) {{1}, {2}, {3, 4, 5}}
(c) {{1, 2, 3, 4}, {5}}