CBSE Class 11 Mathematics Relations and Functions Notes Set 02

Important Notes and Formulas

Cartesian Product of Sets

Let A and B be any two non-empty sets. The set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called the Cartesian product of the sets A and B and is denoted by A × B.
Thus, A × B = {(a, b) : a ∈ A and b ∈ B}.
If A = φ or B = φ, then we define A × B = φ.
Example: Let A = {a, b, c} and B = {p, q}.
Then A × B = {(a, p), (a, q), (b, p), (b, q), (c, p), (c, q)}
Also B × A = {(p, a), (p, b), (p, c), (q, a), (q, b), (q, c)}
If n(A) = p and n(B) = q, then n(A × B) = pq.
In general A × B ≠ B × A, i.e. the Cartesian product of two sets is not commutative.

Important theorems on Cartesian product of sets:
Theorem 1: For any three sets A, B, C: (i) A × (B ∪ C) = (A × B) ∪ (A × C)   (ii) A × (B ∩ C) = (A × B) ∩ (A × C)
Theorem 2: For any three sets A, B, C: A × (B − C) = (A × B) − (A × C)
Theorem 3: If A and B are any two non-empty sets, then A × B = B × A ⇔ A = B.
Theorem 4: If A and B are two non-empty sets having n elements in common, then A × B and B × A have n² elements in common.

Relations

Let A and B be two non-empty sets. Then every subset of A × B defines a relation from A to B, and every relation from A to B is a subset of A × B.
Let R ⊆ A × B and (a, b) ∈ R. Then we say that a is related to b by the relation R and write it as a R b. If (a, b) ∉ R, we say that a is not related to b by R.
Example: Let A = {1, 2, 5, 8, 9}, B = {1, 3}. We set a relation from A to B as: a R b iff a ≤ b; a ∈ A, b ∈ B. Then R = {(1, 1), (1, 3), (2, 3)} ⊆ A × B.

(1) Total number of relations: Let A and B be two non-empty finite sets consisting of m and n elements respectively. Then A × B consists of mn ordered pairs, so the total number of subsets of A × B is 2^mn. Since each subset of A × B defines a relation from A to B, the total number of relations from A to B is 2^mn. Among these 2^mn relations, the void relation φ and the universal relation A × B are trivial relations from A to B.

(2) Domain and range of a relation: Let R be a relation from a set A to a set B. The set of all first components (coordinates) of the ordered pairs belonging to R is called the domain of R, while the set of all second components of the ordered pairs in R is called the range of R.
Thus, Dom (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}.
It is evident from the definition that the domain of a relation from A to B is a subset of A and its range is a subset of B.

(3) Relation on a set: Let A be a non-void set. Then a relation from A to itself, i.e. a subset of A × A, is called a relation on the set A. If n(A) = n, the number of relations on A is 2ⁿ².

Solved Examples

Example 1. If A, B and C are any three sets, then A × (B ∩ C) is equal to
(a) (A × B) ∪ (A × C)
(b) (A × B) ∩ (A × C)
(c) (A ∪ B) × (A ∪ C)
(d) (A ∩ B) × (A ∩ C)
Answer:
Ans. (b) (A × B) ∩ (A × C)
A × (B ∩ C) = (A × B) ∩ (A × C). It is the distributive law.

Example 2. If A, B and C are any three sets, then A × (B ∪ C) is equal to
(a) (A × B) ∪ (A × C)
(b) (A ∪ B) × (A ∪ C)
(c) (A × B) ∩ (A × C)
(d) None of these
Answer:
Ans. (a) (A × B) ∪ (A × C)
It is the distributive law: A × (B ∪ C) = (A × B) ∪ (A × C).

Example 3. If A = {0, 1} and B = {1, 0}, then A × B is equal to
(a) {0, 1, 1, 0}
(b) {(0, 1), (1, 0)}
(c) {0, 0}
(d) {(0, 1), (0, 0), (1, 1), (1, 0)}
Answer:
Ans. (d) {(0, 1), (0, 0), (1, 1), (1, 0)}
By the definition of Cartesian product of sets,
A × B = {(0, 1), (0, 0), (1, 1), (1, 0)}.

Example 4. If A = {2, 4, 5}, B = {7, 8, 9}, then n(A × B) is equal to
(a) 6
(b) 9
(c) 3
(d) 0
Answer:
Ans. (b) 9
A × B = {(2, 7), (2, 8), (2, 9), (4, 7), (4, 8), (4, 9), (5, 7), (5, 8), (5, 9)}
n(A × B) = n(A) · n(B) = 3 × 3 = 9.

Example 5. If the set A has p elements and B has q elements, then the number of elements in A × B is
(a) p + q
(b) p + q + 1
(c) pq
(d) p²
Answer:
Ans. (c) pq
n(A × B) = n(A) · n(B) = pq.

Example 6. If A = {a, b}, B = {c, d}, C = {d, e}, then {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e)} is equal to [AMU 1999; Him. CET 2002]
(a) A ∩ (B ∪ C)
(b) A ∪ (B ∩ C)
(c) A × (B ∪ C)
(d) A × (B ∩ C)
Answer:
Ans. (c) A × (B ∪ C)
B ∪ C = {c, d} ∪ {d, e} = {c, d, e}
∴ A × (B ∪ C) = {a, b} × {c, d, e} = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e)}.

Example 7. If A = {x : x² − 5x + 6 = 0}, B = {2, 4}, C = {4, 5}, then A × (B ∩ C) is [Kerala (Engg.) 2002]
(a) {(2, 4), (3, 4)}
(b) {(4, 2), (4, 3)}
(c) {(2, 4), (3, 4), (4, 4)}
(d) {(2, 2), (3, 3), (4, 4), (5, 5)}
Answer:
Ans. (a) {(2, 4), (3, 4)}
Clearly, A = {2, 3}, B = {2, 4}, C = {4, 5}
B ∩ C = {4}
∴ A × (B ∩ C) = {(2, 4), (3, 4)}.

Example 8. If P, Q and R are subsets of a set A, then R × (P^c ∪ Q^c)^c =
(a) (R × P) ∩ (R × Q)
(b) (R × P) − (R × Q)
(c) (R × P) ∪ (R × Q)
(d) None of these
Answer:
Ans. (a) (R × P) ∩ (R × Q)
R × (P^c ∪ Q^c)^c = R × [(P^c)^c ∩ (Q^c)^c] (De Morgan’s law)
= R × (P ∩ Q) = (R × P) ∩ (R × Q).

Example 9. Let A = {1, 2, 3}. The total number of distinct relations that can be defined over A is
(a) 2⁹
(b) 6
(c) 8
(d) None of these
Answer:
Ans. (a) 2⁹
n(A × A) = n(A) · n(A) = 3² = 9
So, the total number of subsets of A × A is 2⁹, and a subset of A × A is a relation over the set A.

Example 10. Let X = {1, 2, 3, 4, 5} and Y = {1, 3, 5, 7, 9}. Which of the following is/are relations from X to Y
(a) R₁ = {(x, y) | y = 2 + x, x ∈ X, y ∈ Y}
(b) R₂ = {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}
(c) R₃ = {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}
(d) R₄ = {(1, 3), (2, 5), (2, 4), (7, 9)}
Answer:
Ans. (a, b, c) R₁ = {(x, y) | y = 2 + x, x ∈ X, y ∈ Y}; R₂ = {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}; R₃ = {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}
R₁, R₂ and R₃ are subsets of X × Y, so they are relations from X to Y.
R₄ is not a relation from X to Y, because (7, 9) ∈ R₄ but (7, 9) ∉ X × Y.

Example 11. Given two finite sets A and B such that n(A) = 2, n(B) = 3. Then the total number of relations from A to B is
(a) 4
(b) 8
(c) 64
(d) None of these
Answer:
Ans. (c) 64
Here n(A × B) = 2 × 3 = 6
Since every subset of A × B defines a relation from A to B, the number of relations from A to B is equal to the number of subsets of A × B = 2⁶ = 64.

Example 12. The relation R defined on the set of natural numbers as {(a, b) : a exceeds b by 3} is given by
(a) {(1, 4), (2, 5), (3, 6), …}
(b) {(4, 1), (5, 2), (6, 3), …}
(c) {(1, 3), (2, 6), (3, 9), …}
(d) None of these
Answer:
Ans. (b) {(4, 1), (5, 2), (6, 3), …}
R = {(a, b) : a, b ∈ N, a − b = 3} = {((n + 3), n) : n ∈ N} = {(4, 1), (5, 2), (6, 3), …}.