Page 26 - Relations and Functions 19.10.06.pmd

26 MATHEMATICS

Show that ∗ is commutative and associative. Find the identity element for ∗ on
A, if any.
12. State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N, a ∗ a = a ∀ a ∈ N.
(ii) If ∗ is a commutative binary operation on N, then a ∗ (b ∗ c) = (c ∗ b) ∗ a
13. Consider a binary operation ∗ on N defined as a ∗ b = a + b . Choose the
3
3
correct answer.
(A) Is ∗ both associative and commutative?
(B) Is ∗ commutative but not associative?
(C) Is ∗ associative but not commutative?
(D) Is ∗ neither commutative nor associative?

Miscellaneous Examples
Example 41 If R and R are equivalence relations in a set A, show that R ∩ R is

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2
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also an equivalence relation.
Solution Since R and R are equivalence relations, (a, a) ∈ R , and (a, a) ∈ R ∀ a ∈ A.

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This implies that (a, a) ∈ R ∩ R , ∀ a, showing R ∩ R is reflexive. Further,
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2
1
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(a, b) ∈ R ∩ R ⇒ (a, b) ∈ R and (a, b) ∈ R ⇒ (b, a) ∈ R and (b, a) ∈ R ⇒
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1
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1
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(b, a) ∈ R ∩ R , hence, R ∩ R is symmetric. Similarly, (a, b) ∈ R ∩ R and
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(b, c) ∈ R ∩ R ⇒ (a, c) ∈ R and (a, c) ∈ R ⇒ (a, c) ∈ R ∩ R . This shows that
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R ∩ R is transitive. Thus, R ∩ R is an equivalence relation.
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Example 42 Let R be a relation on the set A of ordered pairs of positive integers
defined by (x, y) R (u, v) if and only if xv = yu. Show that R is an equivalence relation.
Solution Clearly, (x, y) R (x, y), ∀ (x, y) ∈ A, since xy = yx. This shows that R is
reflexive. Further, (x, y) R (u, v) ⇒ xv = yu ⇒ uy = vx and hence (u, v) R (x, y). This
shows that R is symmetric. Similarly, (x, y) R (u, v) and (u, v) R (a, b) ⇒ xv = yu and
a a b a
ub = va ⇒ xv = yu ⇒ xv = yu ⇒ xb = ya and hence (x, y) R (a, b). Thus, R
u u v u
is transitive. Thus, R is an equivalence relation.
Example 43 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R be a relation in X given
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by R = {(x, y) : x – y is divisible by 3} and R be another relation on X given by
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R = {(x, y): {x, y} ⊂ {1, 4, 7}} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9}}. Show that
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R = R .
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