Page 27 - Relations and Functions 19.10.06.pmd

RELATIONS AND FUNCTIONS 27

Solution Note that the characteristic of sets {1, 4, 7}, {2, 5, 8} and {3, 6, 9} is
that difference between any two elements of these sets is a multiple of 3. Therefore,
(x, y) ∈ R ⇒ x – y is a multiple of 3 ⇒ {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8}
1
or {x, y} ⊂ {3, 6, 9} ⇒ (x, y) ∈ R . Hence, R ⊂ R . Similarly, {x, y} ∈ R ⇒ {x, y}
1
2
2
2
⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} ⇒ x – y is divisible by
3 ⇒ {x, y} ∈ R . This shows that R ⊂ R . Hence, R = R .
1 2 1 1 2
Example 44 Let f : X → Y be a function. Define a relation R in X given by
R = {(a, b): f(a) = f(b)}. Examine if R is an equivalence relation.
Solution For every a ∈ X, (a, a) ∈ R, since f(a) = f (a), showing that R is reflexive.
Similarly, (a, b) ∈ R ⇒ f(a) = f (b) ⇒ f (b) = f (a) ⇒ (b, a) ∈ R. Therefore, R is
symmetric. Further, (a, b) ∈ R and (b, c) ∈ R ⇒ f (a) = f (b) and f(b) = f (c) ⇒ f (a)
= f(c) ⇒ (a, c) ∈ R, which implies that R is transitive. Hence, R is an equivalence
relation.

Example 45 Determine which of the following binary operations on the set N are
associative and which are commutative.
(ab+ )
(a) a ∗ b = 1 ∀ a, b ∈ N (b) a ∗ b = ∀ a, b ∈ N
2

Solution
(a) Clearly, by definition a ∗ b = b ∗ a = 1, ∀ a, b ∈ N. Also
(a ∗ b) ∗ c = (1 ∗ c) =1 and a ∗ (b ∗ c) = a ∗ (1) = 1, ∀ a, b, c ∈ N. Hence
R is both associative and commutative.
ab+ ba+
(b) a ∗ b = = = b ∗ a, shows that ∗ is commutative. Further,
2 2
⎛ ab+ ⎞
(a ∗ b) ∗ c = ⎜ ⎟ ∗ c.
⎝ 2 ⎠
⎛ ab ⎞ +
⎜ ⎝ 2 ⎠ ⎟ + c ab++ 2c
= = .
2 4
⎛ bc+ ⎞
But a ∗ (b ∗ c) = a ∗ ⎜ ⎟
⎝ 2 ⎠
bc+
a + 2ab c++ a b+ + 2c
= 2 = ≠ in general.
2 4 4
Hence, ∗ is not associative.