# 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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