This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 10 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 10 Solution
10. Give an example of a relation. Which is
i.
Symmetric but neither reflexive nor transitive.
ii.
Transitive but neither reflexive nor symmetric.
iii.
Reflexive and symmetric but not transitive.
iv.
Reflexive and transitive but not symmetric.
v.
Symmetric and transitive but not reflexive.
10.i. Give an example of a relation which is Symmetric but neither reflexive nor transitive.
Consider a set A = {a, b, c} and the relation R defined as R = {(a, b), (b, a)}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
The element (a, a) ∉ R
⇒ This relation is not reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
In this relation, both (a, b) ∈ R and (b, a) ∈ R
⇒ This relation is symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
In this example (a, b) ∈ R and (b, a) ∈ R. But (a, a) ∉ R.
⇒ This relation is not transitive.
∴ R is Symmetric but neither reflexive nor transitive.
10.ii. Give an example of a relation which is Transitive but neither reflexive nor symmetric.
Consider the relation defined by R = {(x, y): x > y}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
As we know, No element greater than itself
⇒ (a, a) ∉ R.
∴ R is not reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
Consider two elements a, b such that {a \gt b}. Obviously {b \ngtr a} (to be specific b \le a).
⇒ (a, b) ∈ R but (b, a) ∉ R.
∴ R is not symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
If there’re 3 elements a, b and c such that {a \gt b} and {b \gt c} then obviously {a \gt c}
⇒ If (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R
⇒ R is transitive.
∴ R is Symmetric but neither reflexive nor transitive.
10.iii. Give an example of a relation which is reflexive and symmetric but not transitive.
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
Consider the set A = {p, q, r}. Consider the relation on the set A = {p, q, r} defined as R = {(p, p), (q, q), (r, r), (p, q), (q, p), (q, r), (r, q)}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
In the set A, we see that For every element a ∈ A, there is a corresponding (a, a) ∈ R.
⇒ (p, p) ∈ R, (q, q) ∈ R and (r, r) ∈ R
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
In the relation R, we see that (p, q) ∈ R and (q, p) ∈ R.
Also (q, r) ∈ R and (r, q) ∈ R
∴ R is symmetric
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
In the relation R, we see that both (p, q) ∈ R and (q, r) ∈ R, but (p, r) ∉ R
∴ R is not transitive.
∴ R is reflexive and symmetric but not transitive.
10.iv. Give an example of a relation which is Reflexive and transitive but not symmetric.
Consider the relation R = {(x, y) : x \geq y} defined on the set of natural numbers N
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
For any natural number a ∈ N, we know that {a = a}.
⇒ a \geq a
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
Consider the element (5, 4) ∈ R because 5 ≥ 4. But we see that {4 \nleq 5}.
⇒ (4, 5) ∉ R
⇒ There are few elements (a, b) ∈ R such that (b, a) ∉ R
∴ R is not symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
When {a \geq b} and {b \geq c}, we have {a \geq c}
⇒ For every (a, b) ∈ R and (b, c) ∈ R, there is a corresponding (a, c) ∈ R
∴ R is transitive.
∴ R is reflexive and transitive but not symmetric.
10.v. Give an example of a relation which is Symmetric and transitive but not reflexive.
Consider the relation R = {(p, q), (q, p), (p, p)} defined on the set A = {p, q}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
We see that (q, q) ∉ R
∴ R is not reflexive
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
We see that (p, q) ∈ R and (q, p) ∈ R
∴ R is symmetric
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
Both (p, q) ∈ R (q, p) ∈ R and also (p, p) ∈ R.
∴ R is not transitive.
∴ R is symmetric and transitive but not reflexive
