Let S = {a,b,c,d} and a relation R={(a,a), (a,c), (a,d), (b,d)} defined on S x S,
then the domain of R is
(A) S
(B) θ
(C) {a,b}
(D) {a,b,c}
Answers
Answer:
9.5 Equivalence Relations
You know from your early study of fractions that each fraction has many equivalent forms. For example,
1
2
,
2
4
,
3
6
,
−1
−2
,
−3
−6
,
15
30
, . . .
are all different ways to represent the same number. They may look different; they may be called different
names; but they are all equal. The idea of grouping together things that “look different but are really the
same” is the central idea of equivalence relations.
A partition of a set S is a finite or infinite collection of nonempty, mutually disjoint subsets whose union
is S.
Definition 1. A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their
union. In other words, the collection of subsets Ai
, i ∈ I (where I is an index set) forms a partition of S if
and only if
(i) Ai 6= ∅ for i ∈ I,
(ii) Ai ∩ Aj = ∅ when i 6= j, and
(iii)
[
i∈I
Ai = S.
(Here the notation S
i∈I
represents the union of the sets Ai for all i ∈ I.)
Definition 2. A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and
transitive.
Recall:
1. R is reflexive if, and only if, ∀x ∈ A, x R x.
2. R is symmetric if, and only if, ∀x, y ∈ A, if x R y then y R x.
3. R is transitive if, and only if, ∀x, y, z ∈ A, if x R y and y R z then x R z.
Definition 3. Two elements a and b that are related by an equivalence relation are called equivalent. The
notation a ∼ b is often used to denote that a and b are equivalent elements with respect to a particular
equivalence relation.
Example 1. Are these equivalence relations on {0, 1, 2}?
(a) {(0, 0),(1, 1),(0, 1),(1, 0)}
(b) {(0, 0),(1, 1),(2, 2),(0, 1),(1, 2)}
(c) {(0, 0),(1, 1),(2, 2),(0, 1),(1, 2),(1, 0),(2, 1)}
(d) {(0, 0),(1, 1),(2, 2),(0, 1),(0, 2),(1, 0),(1, 2),(2, 0),(2, 1)}
(e) {(0, 0),(1, 1),(2, 2)}
Solution. (a) R is not reflexive: (2, 2) ∈/ R. Thus, by definition, R is not an equivalence relation.
(b) R is not symmetric: (1, 2) ∈ R but (2, 1) ∈/ R. Thus R is not an equivalence relation.
(c) R is not transitive: (0, 1),(1, 2) ∈ R, but (0, 2) ∈/ R. Thus R is not an equivalence relation.
(d) R is reflexive, symmetric, and transitive. Thus R is an equivalence relation.
Step-by-step explanation:
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