12. If A and B be non-empty sets and A × B= A × C then show that B=C - Advance Math Class 10 - Exercise: 1.2

12. If A and B be non-empty sets and A × B= A × C then show that B=C - Advance Math Class 10 - Exercise: 1.2

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Solution:

⇒ Let y be an arbitrary element of B

⇒ Therefore, 

⇒ (x, y) Є A×B (ꓯ x Є A)

⇒ (x, y) Є A×C ( A×B = A×C)

⇒ y Є C                 …….(1)

⇒Again, Let x be any arbitrary element of C

⇒Therefore, x Є C

⇒ (y, x) Є A×C (ꓯ y Є A)

⇒ (y, x) Є A ×B (A×C= A ×B)

⇒ X Є B                ……(2)

⇒ From (1) and (2), we get

⇒ B = C

  Exercise: 1.2  

1. If A = {1, 3, 5} and B= {2, 4, -1} then find A×B and B×A . Find the number of elements in A×B and B×A.

2. (a) If A = {-1, 3, 6} and B= {-3, 5} then find A×B and B×A. Also draw the graph of the Cartesian products.

(b) If A= {2, 4}, B= {-1, 3, 7}, C={1, 0} then with the help of tree diagram find A×B, B×A and A×B×C.

3. If A= {x, y, u, v} and B= {a, b, l, m} then find A×B and represent the Cartesian product by a coordination diagram.

4. If A= {3, 5}, B= {1, 2, 4}, C= {3, 4, 6} then show that

(i) A×(BUC) = (A×B) U (A×C)

(ii) A×(BꓵC) = (A×B) ꓵ (A×C)

5. For the set A= {x:x Є I and -2 < x < 4} and B= {y:y is a root of y2 – 9=0} draw the graph of A×B and find n(A×B)

6. Draw the graphs of A×B and B×A on two separate coordinate planes for the set A={-3, 0, 3} and B= {-1, 1}

7. (i) If A=ф and B={-1, 1} then determine the power set of A×B, B×A and B2

(ii ) If A= {0} and B= {1} then determine the A×B and B×A. Also determine P(A×B) and P(B×A)

8. If A= {2, 4} and B = {4, 2}, Can we write A×B = A2 and B×A= B2 ?

9. If n(A) = 3 and two of the elements of A × A be (a, a), (b, c) then determine the sets A and A × A.

10. If A ⊆ B and C ⊆ D then show that A×C ⊆ B×D

11. If A×B= {(m, 1), (n, 3), (m, 3), (n, 1), (m, 2), (n, 2)}, then find A and B.

12. If A and B be non-empty sets and A × B= A × C then show that B=C.

13. A= {1, 2}, B= {1, 3, 2} and C= {4, 2, 1} then show that A2 = B2ꓵC2

14. If A={0, 1} then find A×A×A