Mathematics Lab Activity-5 Class XI

Mathematics Laboratory Activities on Relaton & Functons for class XI students Non-Medical. These activities are strictly according to the CBSE syllabus

Chapter - 2 | relations & functions

Activity - 5

Objective

To verify that for two sets A and B, n(A x B) = pq and the total number of relations from A to B is 2^pq, where n(A) = p and n(B) = q

Material Required

Paper , Different colored pencils.                                                                                  

Theory

Cartesian Product : If A and B are two non-empty sets, then the Cartesian product  A x B is defined as the set of all the ordered pairs of the elements from A to B such that 

A x B = {(a, b) : a ∈ A, b ∈ B}

First element of all the ordered pair ∈ set A and the second element ∈ set B

For Example:

 If A = {5, 6} and B = {2, 3, 4}, then A X B = {(5, 2), (5,  3),(5, 4), (6, 2), (6, 3), (6, 4)}

Number of elements in the set A, n(A) = 2

Number of elements in the set B, n(B) = 3

Number of elements in the set A X B,  n(A X B) = 6

⇒ n(A x B) = n(A) x n(B)

Relations: If A and B are two non-empty sets, then relation R from A to B is a subset of the Cartesian product A x B. This means that number of subsets is equal to the number of relations.

Procedure

 1) Take a set A₁ which has one element a₁(say), and take another set B₁, which has one element b₁ (say). Relation is {(a₁, b₁), (b₁, a₁)}

 Represent all possible correspondences of the elements of set A₁ to the elements of set B₁ as shown in figure 2.1

2) Take a set A₂ which has two element {a_1, a₂} (say), and take another set B₃, which has three element {b_1, b₂, b₃ } (say). Relation is {(a₁, b₁), (a₁, b₂), (a₁, b₃), (a₂, b₁), (a₂, b₂), (a₂, b₃)}

Represent all possible correspondences of the elements of set A₂ to the elements of set B₃ as shown in figure 2.2

3) Take a set A₃ which has three element a_1, a_2, a₃ (say), and take another set B₄, which has four element b_1, b₂, b_3, b₄  (say). Relation is {(a₁, b₁), (a₁, b₂), (a₁, b₃), (a1, b4),  (a₂, b₁), (a₂, b₂), (a₂, b₃), (a₂, b₄), (a₃, b₁), (a₃, b₂), (a₃, b₃), (a₃, b₄)}

Represent all possible correspondences of the elements of set A₂ to the elements of set B₃ as shown in figure 2.3

4) Similar visual representations can be shown with different number of elements of two given sets A and B.

Observations

1) The number of arrows in fig 2.1 = 1

 Number of elements in Cartesian product  (A₁ x B₁) of set A₁ and B₁ = 1 x 1=1

No of relations from A₁ to B₁ = 2^1x1  = 2¹

2) The number of arrows in fig 2.2 = 6

 Number of elements in Cartesian product (A₂ x B₃) of set A₂ and B₃ = 2 x 3 = 6

No of relations from A₂ to B₃ = 2^2 x 3 = 2⁶

3) The number of arrows in fig 2.3 = 12

 Number of elements in Cartesian product (A₃ x B₄) of set A₃ and B₄ = 3 x 4 = 12

No of relations from A₃ to B₄ = 2^3 x 4 = 2¹²

Result

Therefore the number relations from A to B = Number of subsets from A to B = 2^pq   where pq is the number of elements in A x B.

Applications

This activity is used to find the number of subsets of the Cartesian product A x B and the number of relations from A to B.

VIVA – VOICE

Q. 1. Define Cartesian product of two sets.

Ans. If A and B are two non-empty sets, then the Cartesian product  A x B is defined as the set of all the ordered pairs of the elements from A to B such that 

A x B = {(a, b) : a ∈ A, b ∈ B}

First element of all the ordered pair ∈ set A and the second element ∈ set B

For Example: If A = {5, 6} and B = {2, 3, 4}, then A X B = {(5, 2), (5, 3), (5, 4), (6, 2), (6, 3), (6, 4)}

Note: If either A or B is a null set , then A x B will also be a null set.

Q. 2. If n(A) = p and n(B) = q, then what is n(A x B).

Ans. n(A x B) = pq

Q. 3. Define a relation.

Ans. A relation R from set A to set B is the subset Cartesian product A x B.

In Cartesian product A x B number of relations equal to the number of subsets.

Q. 4. What is the  domain of a relation R.

Ans. Set of all first elements of the ordered pairs in the relation R is called its domain.

Q. 5. What is the range of a relation R.

Ans. Set of all second elements of the ordered pairs in the relation R is called its range.

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