Let A = {1, 2, 3, 4, 5, 6}
R = {(a, b) : b = a + 1} = {(a, a + 1)}
= {(1, 2), (2, 3), (3, 4), (4,5), (5,6)}
(i) R is not reflexive as (a, a) ∉ R ∀ a ∈ A
(ii) (a,b) ∈ R ⇏ (b,a) ∈ R [∵ (a, b) ∈ R ⇒ b = a + 1 ⇒ a = b –1]
∴ R is not symmetric.
(iii) (a, b) ∈ R, (b, c) ∈ R ⇏ (a, c) ∈ R
[∵ (a, b), (b, c) ∈ R ⇒ b = a + 1, c = b + 1 ⇒ c = a + 2]
∴ R is not transitive.
R = {(a, b) : a ≤ b3}
(i) Since (a, a) ∉ R as a ≤ a3 is not always true
[Take a = 1/3. then a ≤ a3 is not true]
∴ R is not reflexive.
(ii) Also (a, b) ∈ R ⇏ (b, a) ∈ R
[Take a = 1, b = 4, ∴ 1 ≤ 43 but 4 ≰ (l)3 ]
∴ R is not symmetric.
(iii) Now (a, b) ∈ R, (b, c) ∈ R ⇏ (a, c) ∴ R
[Take a = 100, b = 5, c = 3, ∴ 100 ≤ 53, 5 ≤ 33 but 100 ≥ 33] R is not symmetric.
R = {(a, b) : a ≤ b}
(i) Since (a, a) ∈ R ∀ a ∈ R [∵ a ≤ a ∀ a ∈ R]
∴ R is reflexive.
(ii) (a, b) ∈ R ⇏ (b, a) ∈ R [∵ if a ≤ b. then b ≤ a is not true]
∴ R is not symmetric.
(iii) Let (a, b), (b, c) ∈ R ∴ a ≤ b, b ≤ c ∴ a ≤ c ⇒ (a, c) ∈ R ∴ (a, b), (b. c) ∈ R ⇒ (a, c) ∈ R ∴ R is transitive
Determine whether each of the following relations are reflexive, symmetric and transitive :
(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as
R = {(x, y) : 3 x – y = 0}
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y) : x and y work at the same place}
(b) R = {(x,y) : x and y live in the same locality}
(c) R = {(x, y) : x is exactly 7 cm taller than y}
(d) R = {(x, y) : x is wife of y}
(e) R = {(x,y) : x is father of y}
(i) A = {1,2,3,.....,13,14}
R = {x.y) : 3 x – y ≠} = {(x, y) : y = 3 x}
= {(1,3), (2, 6), (3, 9), (4, 12)}
(a) R is not reflexive as (x, x) ∉ R [ ∵ 3 x – x ≠ 0]
(b) R is not symmetric as (x,y) ∈ R does not imply (y, x) ∈ R
[ ∴ (1, 3) ∈ R does not imply (3. 1) ∈ R]
(c) R is not transitive as (1.3) ∈ R , (3, 9) ∈ R but (1.9) ∉ R.
(ii) Relation R is in the set N given by
R = {(x, y) : y = x + 5 and x < 4 }
∴ R = {(1,6), (2, 7). (3, 8)}
(a) R is not reflexive as (x, x) ∉ R (b) R is not symmetric as (x, y) ∈ R ⇏ (v, x) ∈ R (c) R is not transitive as (x,y) ∈ R, (y, z) ∈ R ⇏ (x, z) ∈ R
(iii) A = {1, 2, 3, 4, 5, 6}
R = {(x, y) : y is divisible by x}
(a) R is reflexive as (x, x) ∈ R ∀ x ∈ A [∴ x divides x ∀ x ∈ A]
(b) R is not symmetric as (1, 6) ∈ R but (6, 1) ∉ R.
(c) Let (x, y), (y, z) ∈ A
∴ y is divisible by x and z is divisible by y ∴ z is divisible by x
∴ (r, y) ∈ R (y, z) ∈ R ⇒ (x, z) ∈ R ∴ R is transitive.
(iv) Relation R is in the set Z given by R = {(x,y) : x – y is an integer} (a) R is reflexive as ( x, x) ∈ R [∴ x – x = 0 is an integer]
(b) R is symmetric as (x,y) ∈ R ⇒ (y, x) ∈ A
[∵ x – y is an integer ⇒ y – x is an integer]
(c) R is transitive as (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R
[∵ if x – y, y – z are integers, then (x – y) + (y – z) = x – z is also in integer]
(v) A is the set of human beings in a town at a particular time R is relation in A.
(a) R = {(x, y) : x and y work at the same time}
R is reflexive as (x, x) ∈ R R is symmetric as ( x, y) ∈ R ∈ (y, x) ∈ R
[ ∵ x and y work at the same time ⇒ y and x work at the same time] R is transitive as (x, y), (y, z) ∈ R ⇒ (x, z) ∈ R
[∴ if x and y, y and z work at the same time, then x and z also work at the same time]
(b) R = {(x,y) : x and y live in the same locality}
R is reflexive as (x, x) ∈ R R is symmetric as ( x, y) ∈ R ⇒ (y, x) ∈ R
[∴ x and y live in the same locality ⇒ y and x live in the same locality] R is transitive as ( x, y), ( y, z) ∈ R ⇒ (x, z) ∈ R
[∵ if x and y, y and z live in the same locality. then x and z also live in the same locality]
(c) R = {(x,y) : x is exactly 7 cm taller than y}
Since (x, x) ∉ R as x cannot be 7 cm taller than x.
∴ R is not reflexive.
(x, y) ∈ R ⇒ (y.x) ∈ R as if x is taller than y, then y cannot be taller than x.
∴ R is not symmetric.
Again (x,y), (y,z) ∈ R ⇏ (x, z) ∈ R
[∵ if x is taller than y by 7 cm and y is taller than z by 7 cm,
then x is taller than z by 14 cm]
∴ R is not transitive.
(d) R = {(x,y) : x is wife of y}
R is not reflexive as (x,y) ∉ R [∴ x cannot be wife of x]
Also (x, y) ∈ R ⇏ (y, x) ∈ R [∵ if x is wife of y, then y cannot be wife of x] ∴ R is not symmetric
R is not transitive.
(e) R = {(x,y) : x is father of y}
R is not reflexive as (x, x) ∉ R [ ∵ x cannot be father of x]
Also (x,y) ∈ R ⇐ (y, x) ∈ R [ ∵ if x is father of y. then y cannot be father of x] ∴ R is not symmetric.
R is not transitive.
R = {(a, b) : a ≤ b2}
(i) Since (a, a) ∉ R
∴ R is not reflexive.
(ii) Also (a, b) ∈ R ⇏ (b, a) ∈ R
[Take a = 2 ,b = 6, then 2 ≤ 62 but (6)2 < 2 is not true]
∴ R is not symmetric.
(iii) Now (a, b), (b, c) ∈ R ∉ (a, c) ∈ R
[Take a = 1, b = – 2, c = – 3 ∴ a ≤ b2 . b ≤ c2 but a ≤ c2 is not true) ∴ R is not transitive.
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