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) : 3x – 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}
Solution:
(i) R = {(1, 3), (2, 6), (3, 9), (4, 12)}
R is not reflexive because (1, 1), (2, 2) ... and (14, 14) ∉ R .
R is not symmetric because (1, 3) ∈ R, but (3, 1) ∉ R. [since 3(3) ≠ 0].
R is not transitive because (1, 3), (3, 9) ∈ R, but (1, 9) ∉ R. [3(1) - 9 ≠ 0].
Hence, R is neither reflexive nor symmetric nor transitive.
(ii) R = {(1, 6), (2, 7), (3,8)}
R is not reflexive because (1, 1) ∉ R .
R is not symmetric because (1, 6) ∈ R but (6,1) ∉ R .
R is not transitive because there isn’t any ordered pair in R such that (x, y), (y, z) ∈ R, so ( x, z ) ∉ R.
Hence, R is neither reflexive nor symmetric nor transitive.
(iii) R = {(x, y) : y is divisible by x}
We know that any number other than 0 is divisible by itself.
Thus, (x, x) ∈ R
So, R is reflexive.
(2, 4) ∈ R [because 4 is divisible by 2]
But (4, 2) ∉ R [since 2 is not divisible by 4]
So, R is not symmetric.
Let (x, y) and (y, z) ∈ R. So, y is divisible by x and z is divisible by y.
So, z is divisible by x ⇒ (x, z) ∈ R
So, R is transitive.
So, R is reflexive and transitive but not symmetric.
(iv) R = {(x, y) : x - y is an integer}
For x ∈ Z, (x, x) ∉ R because x - x = 0 is an integer.
So, R is reflexive.
For, x, y ∈ Z , if x, y ∈ R , then x - y is an integer ⇒ (y - x) is an integer.
So, ( y, x) ∈ R
So, R is symmetric. Let (x, y) and (y, z) ∈ R, where x, y, z ∈ Z.
⇒ (x - y) and (y - z) are integers.
⇒ x - z = (x - y) + (y - z) is an integer. So, R is transitive.
So, R is reflexive, symmetric, and transitive.
(v)
(a) R = {(x, y) : x and y work at the same place}
R is reflexive because ( x, x) ∈ R
R is symmetric because ,
If (x, y) ∈ R, then x and y work at the same place, and y and x also work at the same place. (y, x) ∈ R. R is transitive because,
Let (x, y), (y, z) ∈ R
x and y work at the same place and y and z work at the same place.
Then, x and z also work at the same place. (x, z) ∈ R.
Hence, R is reflexive, symmetric, and transitive.
(b) R = {(x, y) : x and y live in the same locality}
R is reflexive because (x, x) Î R
R is symmetric because,
If ( x, y) ∈ R, then x and y live in the same locality and y and x also live in
the same locality (y, x) ∈ R. R is transitive because,
Let (x, y), (y, z ) ∈ R
x and y live in the same locality and y and z live in the same locality.
Then x and z also live in the same locality. (x, z) ∈ R.
Hence, R is reflexive, symmetric, and transitive.
(c) R = {(x, y) : x is exactly 7cm taller than y}
R is not reflexive because (x, x) ∉ R.
R is not symmetric because,
If (x, y) ∈ R, then x is exactly 7cm taller than y, and y is clearly not taller than x. (y, x) ∉ R.
R is not transitive because Let (x, y), (y, z) ∈ R
x is exactly 7cm taller than y and y is exactly 7cm taller than z.
Then x is exactly 14cm taller than z. (x, z) ∉ R.
Hence, R is neither reflexive nor symmetric nor transitive.
(d) R = {(x, y ) : x is wife of y}
R is not reflexive because (x, x) ∉ R.
R is not symmetric because,
Let (x, y) ∈ R, x is the wife of y and y is not the wife of x. (y, x) ∉ R.
R is not transitive because,
Let (x, y), (y, z) ∈ R.
x is the wife of y and y is the wife of z, which is not possible.
(x, z) ∈ R.
Hence, R is neither reflexive nor symmetric nor transitive.
(e) R = {(x, y) : x is father of y}
R is not reflexive because (x, x) ∉ R.
R is not symmetric because,
Let ( x, y) ∈ R, x is the father of y and y is not the father of x. (y, x) ∉ R.
R is not transitive because,
Let ( x, y), (y, z) ∈ R
x is the father of y and y is the father of z, x is not the father of z. (x, z) ∉ R.
Hence, R is neither reflexive nor symmetric nor transitive
NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 1
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) : 3x – 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}
Summary:
From the given set of relations, we have obtained whether each of the following relations is reflexive, symmetric, and transitive. A relation that shows all three properties are called an equivalence relation
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