Relations and Functions
QUICK RECAP
RELATION of A, then relation R on A is called an empty
relation i.e., R = f ⊂ A × A.
8 A relation R from a set A to a set B is a subset
of A × B. So, we say R ⊆ A × B. A relation Universal Relation
from a set A to itself is called a relation in A. X If each element of A is related to every
Empty Relation element of A, then relation R in A is called
X If no element of A is related to any element universal relation i.e., R = A × A.
8 A relation R in a set A is called distinct elements of A have distinct
(i) reflexive, if (a, a) ∈ R, for all a ∈ A images in B i.e., for a, b ∈ A, f(a) = f(b)
(ii) symmetric, if (a, b) ∈ R ⇒ (b, a) ∈ R, ⇒a=b
for all a, b ∈ A (ii) onto or surjective function, if for every
(iii) transitive, if (a, b) ∈ R and (b, c) ∈ R element b ∈ B, there exists some a ∈ A
⇒ (a, c) ∈ R, for all a, b, c ∈ A such that f(a) = b.
X A relation R in a set A is called an equivalence X A function f : A → B is called bijective
relation, if it is reflexive, symmetric and function, if it is both one-one and onto
transitive. function.
X In a relation R in a set A, the set of all elements Composition of Functions
related to any element a ∈ A is denoted by
[a] i.e., [a] = {x ∈ A : (x, a) ∈ R} X Let f : A → B and g : B → C be any two functions,
Here, [a] is called an equivalence class of then the function gof : A → C defined as
a ∈ A. gof(x) = g(f(x)), for all x ∈ A, is called the
composition of f and g.
FUNCTION
Invertible Functions
8 A relation f from a set A to a set B is called a
X A function f : A → B is said to be invertible,
function if
(i) for each a ∈ A, there exists some b ∈ B if there exists a function g : B → A such that
such that (a, b) ∈ f i.e., f(a) = b gof = IA and fog = IB. Here, g is called the
(ii) (a, b) ∈ f and (a, c) ∈ f ⇒ b = c inverse of f.
X A function f : A → B is called X Also, f is an invertible function iff it is a
(i) one-one or injective function, if bijective function.
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
1.2 Types of Relations 11. Show t hat t he rel at i on S i n t he s e t
A = {x ∈ Z : 0 ≤ x ≤ 12} given by
VSA (1 mark) S = {(a, b) : a, b ∈ Z, |a – b| is divisible by 3}
1. A relation R in a set A is called , if is an equivalence relation. (AI 2019)
(a 1, a 2) ∈ R implies (a 2, a 1) ∈ R, for all 12. Let A = {1, 2, 3, ..., 9} and R be the relation in
a1, a2 ∈ A. (2020) A × A defined by (a, b) R (c, d) if a + d = b + c
2. A relation in a set A is called for (a, b), (c, d) in A × A. Prove that R is
relation, if each element of A is related to an equivalence relation. Also obtain the
itself. (2020) equivalence class [(2, 5)]. (Delhi 2014)
3. If R = {(x, y) : x + 2y = 8} is a relation on N, 13. Let R be a relation defined on the set of
write the range of R. (AI 2014) natural numbers N as follow :
R = {(x, y) | x ∈ N, y ∈ N and 2x + y = 24}
4. Let R = {(a, a3) : a is a prime number less
Find the domain and range of the relation R.
than 5} be a relation. Find the range of R.
Also, find if R is an equivalence relation or
(Foreign 2014)
not. (Delhi 2014 C)
5. Let R be the equivalence relation in the set
A = {0, 1, 2, 3, 4, 5} given by LA 2 (6 marks)
R = {(a, b) : 2 divides (a – b)}. Write the 14. Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that
equivalence class [0]. (Delhi 2014 C)
R = {(a, b) : a, b ∈ A, |a – b| is divisible
6. State the reason for the relation R in the set by 4}, is an equivalence relation. Find the
{1, 2, 3} given by R = {(1, 2), (2, 1)} not to be set of all elements related to 1. Also write the
transitive. (Delhi 2011) equivalence class [2]. (2018)
SA (2 marks) 15. Let N denote the set of all natural numbers
and R be the relation on N × N defined by
7. Check if the relation R in the set of real (a, b) R(c, d) if ad(b + c) = bc(a + d). Show
numbers defined as R = {(a, b) : a < b} is that R is an equivalence relation.
(i) symmetric, (ii) transitive. (2020) (Delhi 2015)
LA 1 (4 marks) 16. Show that the relation R in the set A = {1, 2,
3, 4, 5} given by R = {(a, b) : |a – b| is divisible
8. Let N be the set of natural numbers and R
by 2} is an equivalence relation. Write all the
be the relation on N × N defined by (a, b) R
equivalence classes of R. (AI 2015 C)
(c, d) iff ad = bc for all a, b, c, d ∈ N. Show
that R is an equivalence relation. (2020)
1.3 Types of Functions
9. Show that the relation R in the set A = {1, 2, 3,
4, 5, 6} given by R = {(a, b) : |a – b| is divisible VSA (1 mark)
by 2} is an equivalence relation. (2020) 17. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let
10. Show that the relation R on defined as f = {(1, 4), (2, 5), (3, 6)} be a function from A
R = {(a, b) : a ≤ b}, is reflexive and transitive to B, state whether f is one-one or not.
but not symmetric. (Delhi 2019) (AI 2011)
QUICK RECAP
RELATION of A, then relation R on A is called an empty
relation i.e., R = f ⊂ A × A.
8 A relation R from a set A to a set B is a subset
of A × B. So, we say R ⊆ A × B. A relation Universal Relation
from a set A to itself is called a relation in A. X If each element of A is related to every
Empty Relation element of A, then relation R in A is called
X If no element of A is related to any element universal relation i.e., R = A × A.
8 A relation R in a set A is called distinct elements of A have distinct
(i) reflexive, if (a, a) ∈ R, for all a ∈ A images in B i.e., for a, b ∈ A, f(a) = f(b)
(ii) symmetric, if (a, b) ∈ R ⇒ (b, a) ∈ R, ⇒a=b
for all a, b ∈ A (ii) onto or surjective function, if for every
(iii) transitive, if (a, b) ∈ R and (b, c) ∈ R element b ∈ B, there exists some a ∈ A
⇒ (a, c) ∈ R, for all a, b, c ∈ A such that f(a) = b.
X A relation R in a set A is called an equivalence X A function f : A → B is called bijective
relation, if it is reflexive, symmetric and function, if it is both one-one and onto
transitive. function.
X In a relation R in a set A, the set of all elements Composition of Functions
related to any element a ∈ A is denoted by
[a] i.e., [a] = {x ∈ A : (x, a) ∈ R} X Let f : A → B and g : B → C be any two functions,
Here, [a] is called an equivalence class of then the function gof : A → C defined as
a ∈ A. gof(x) = g(f(x)), for all x ∈ A, is called the
composition of f and g.
FUNCTION
Invertible Functions
8 A relation f from a set A to a set B is called a
X A function f : A → B is said to be invertible,
function if
(i) for each a ∈ A, there exists some b ∈ B if there exists a function g : B → A such that
such that (a, b) ∈ f i.e., f(a) = b gof = IA and fog = IB. Here, g is called the
(ii) (a, b) ∈ f and (a, c) ∈ f ⇒ b = c inverse of f.
X A function f : A → B is called X Also, f is an invertible function iff it is a
(i) one-one or injective function, if bijective function.
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
1.2 Types of Relations 11. Show t hat t he rel at i on S i n t he s e t
A = {x ∈ Z : 0 ≤ x ≤ 12} given by
VSA (1 mark) S = {(a, b) : a, b ∈ Z, |a – b| is divisible by 3}
1. A relation R in a set A is called , if is an equivalence relation. (AI 2019)
(a 1, a 2) ∈ R implies (a 2, a 1) ∈ R, for all 12. Let A = {1, 2, 3, ..., 9} and R be the relation in
a1, a2 ∈ A. (2020) A × A defined by (a, b) R (c, d) if a + d = b + c
2. A relation in a set A is called for (a, b), (c, d) in A × A. Prove that R is
relation, if each element of A is related to an equivalence relation. Also obtain the
itself. (2020) equivalence class [(2, 5)]. (Delhi 2014)
3. If R = {(x, y) : x + 2y = 8} is a relation on N, 13. Let R be a relation defined on the set of
write the range of R. (AI 2014) natural numbers N as follow :
R = {(x, y) | x ∈ N, y ∈ N and 2x + y = 24}
4. Let R = {(a, a3) : a is a prime number less
Find the domain and range of the relation R.
than 5} be a relation. Find the range of R.
Also, find if R is an equivalence relation or
(Foreign 2014)
not. (Delhi 2014 C)
5. Let R be the equivalence relation in the set
A = {0, 1, 2, 3, 4, 5} given by LA 2 (6 marks)
R = {(a, b) : 2 divides (a – b)}. Write the 14. Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that
equivalence class [0]. (Delhi 2014 C)
R = {(a, b) : a, b ∈ A, |a – b| is divisible
6. State the reason for the relation R in the set by 4}, is an equivalence relation. Find the
{1, 2, 3} given by R = {(1, 2), (2, 1)} not to be set of all elements related to 1. Also write the
transitive. (Delhi 2011) equivalence class [2]. (2018)
SA (2 marks) 15. Let N denote the set of all natural numbers
and R be the relation on N × N defined by
7. Check if the relation R in the set of real (a, b) R(c, d) if ad(b + c) = bc(a + d). Show
numbers defined as R = {(a, b) : a < b} is that R is an equivalence relation.
(i) symmetric, (ii) transitive. (2020) (Delhi 2015)
LA 1 (4 marks) 16. Show that the relation R in the set A = {1, 2,
3, 4, 5} given by R = {(a, b) : |a – b| is divisible
8. Let N be the set of natural numbers and R
by 2} is an equivalence relation. Write all the
be the relation on N × N defined by (a, b) R
equivalence classes of R. (AI 2015 C)
(c, d) iff ad = bc for all a, b, c, d ∈ N. Show
that R is an equivalence relation. (2020)
1.3 Types of Functions
9. Show that the relation R in the set A = {1, 2, 3,
4, 5, 6} given by R = {(a, b) : |a – b| is divisible VSA (1 mark)
by 2} is an equivalence relation. (2020) 17. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let
10. Show that the relation R on defined as f = {(1, 4), (2, 5), (3, 6)} be a function from A
R = {(a, b) : a ≤ b}, is reflexive and transitive to B, state whether f is one-one or not.
but not symmetric. (Delhi 2019) (AI 2011)
