Page 21 - Mathematics Class - XI

OBSERVATION
1. Cartesian product (A × B ) = {a } × {b } = {a ,b }
1 1 1 1 1 1
n (A × B ) = 1 = 1 × 1 = n (A ) × n (B )
1 1 1 1
Therefore, total number of relations from A to B is 2 .
1
1 1
2. Cartesian product (A × B ) = {a , a } × {b , b , b } = {(a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b )}
1
2
1
1
1
1
2
1
2
3
2
2
3
1
2
3
2
2
2
n (A × B ) = 6 = 2 × 3 = n (A ) × n (B )
2 2 2 2
Therefore, total number of relations from A to B is 2 .
6
2
2
3. Cartesian product (A × B ) = {a , a , a } × {b , b , b , b } = {(a , b ), (a , b ), (a , b ), (a , b ), (a , b ),
1
2
1
3
2
1
1
1
1
1
2
4
1
4
2
3
3
3
3
(a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b )}
2 2 2 3 2 4 3 1 3 2 3 3 3 4
n (A × B ) = 12 = 3 × 4 = n (A ) × n (B )
3
3
3
3
Therefore, total number of relations from A to B is 2 .
12
3
3
4. Cartesian product (A × B ) = {a , a , a , a } × {b , b , b , b , b }
4 4 1 2 3 4 1 2 3 4 5
= {(a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ),
2
1
4
1
2
1
2
1
2
4
3
2
5
1
2
3
1
1
5
2
2
1
3
3
(a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b ), (a , b )}
3
3
3
5
4
3
4
1
4
4
2
4
3
4
5
4
n (A × B ) = 20 = 4 × 5 = n (A ) × n (B )
4 4 4 4
Therefore, total number of relations from A to B is 2 .
20
4
4
CONCLUSION
This activity verifies that for two non empty sets A and B, n (A × B) = n (A) × n (B) and the total number of
relations from A to B is 2 n(A × B) .
APPLICATION
This activity can be used to find the total number of relations from any set A to another set B.
Knowledge Booster
● If A and B are two finite sets, then n(A × B) = n(A) × n(B).
● If either A or B is an infinite set, then A × B is an infinite set.
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