Class 12 Maths – Chapter 1
Relations and Functions
Exercise 1.1 (Complete Solution)
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} defined 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) x and y work at the same place
(b) x and y live in the same locality
(c) x is exactly 7 cm taller than y
(d) x is wife of y
(e) x is father of y
Solution
(i)
Given: R = {(x, y) : 3x − y = 0}
- Reflexive: For reflexive, (x, x) ∈ R ⇒ 3x − x = 0 ⇒ 2x = 0,
which is not true for all x ∈ A.
❌ Not Reflexive - Symmetric: If (x, y) ∈ R ⇒ y = 3x.
But (y, x) ⇒ x = 3y, which is not true.
❌ Not Symmetric - Transitive: If (x, y) and (y, z) ∈ R,
then y = 3x and z = 3y ⇒ z = 9x, which is not in R.
❌ Not Transitive
Conclusion: Relation R is neither reflexive nor symmetric nor transitive.
(ii)
Given: R = {(x, y) : y = x + 5 and x < 4}
- Reflexive: y = x + 5 ⇒ x ≠ y.
❌ Not Reflexive - Symmetric: If y = x + 5, then x ≠ y + 5.
❌ Not Symmetric - Transitive: x → y → z gives z = x + 10,
which is not in R.
❌ Not Transitive
Conclusion: Relation R is neither reflexive nor symmetric nor transitive.
(iii)
Given: R = {(x, y) : y is divisible by x}
- Reflexive: Every number divides itself.
✔ Reflexive - Symmetric: 2 divides 4 but 4 does not divide 2.
❌ Not Symmetric - Transitive: If x divides y and y divides z, then x divides z.
✔ Transitive
Conclusion: Relation R is reflexive and transitive but not symmetric.
(iv)
Given: R = {(x, y) : x − y is an integer}
- Reflexive: x − x = 0, which is an integer.
✔ Reflexive - Symmetric: If x − y is integer, then y − x is also integer.
✔ Symmetric - Transitive: If x − y and y − z are integers, then x − z is integer.
✔ Transitive
Conclusion: Relation R is reflexive, symmetric and transitive.
(v)
(a) x and y work at the same place
✔ Reflexive ✔ Symmetric ✔ Transitive
(b) x and y live in the same locality
✔ Reflexive ✔ Symmetric ✔ Transitive
(c) x is exactly 7 cm taller than y
❌ Reflexive ❌ Symmetric ❌ Transitive
(d) x is wife of y
❌ Reflexive ❌ Symmetric ❌ Transitive
(e) x is father of y
❌ Reflexive ❌ Symmetric ❌ Transitive
