Exercise 1.1.
( 2) Let A = {1, 2, 3} and B{ x | x is a prime number less than 10}. Find A x B and B X A
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Q : 3 . If B X A = { (-2, 3), (-2, 4) (0, 3), (0, 4), (3, 3), (3,4) } find A and B
To determine sets A and B from the given Cartesian product B×A,
we analyze the pairs in the set { (−2, 3), (−2, 4), (0, 3), (0, 4), (3, 3), (3, 4)}.
The Cartesian product B×A is formed by pairing each element of set B with each element of set A. Therefore, we need to extract the distinct elements of A and B from these pairs.
First, we identify the elements of B by examining the first component of each pair: B={−2, 0, 3}
Next, we identify the elements of A by examining the second component of each pair: A={3, 4}
Therefore, the sets A and B are: A={3, 4 B={−2, 0, 3}
If you can't read the text verify the below Image:
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Q : 4 ) If A = {5, 6}, B={4, 5, 6}, C ={5, 6, 7}, Show that A X A = (B x B) ∩ (C X C)
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Question 6: Let A = {x ∈ W | x < 2} , B = {x ∈ N | 1 < x ≤ 4} and C = {3, 5} . Verify that
(i) A × (B U C) = (A × B) U (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
(iii) (A U B) × C = (A × C) U (B × C)
Solution:
A = {x ∈ W | x < 2} , B = {x ∈|1 < x ≤ 4} and C = {3, 5} .
A = {0, 1} , B = {2, 3, 4} and C = {3, 5} .
(i) A × (B U C) = (A × B) U (A × C)
Consider the LHS
(B U C) = {2, 3, 4, 5}
A × (B U C)
= {(0, 2) (0, 3) (0, 4) (0, 5)(1, 2) (1, 3) (1, 4) (1, 5)} ——-(1)
Take the RHS
(A × B) = {(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)}
(A × C) = {(0, 3)(0, 5) (1, 3) (1, 5)}
(A x B) U (A x C) = {(0, 2) (0, 3) (0, 4) (0, 5)(1, 2) (1, 3) (1, 4) (1, 5)} ——(2)
(1) = (2)
(ii) A × (B n C) = (A × B) n (A × C)
A = {0, 1} , B = {2, 3, 4} and C = {3, 5} .
Consider the LHS
(B n C) = {3}
A × (B n C) = {(0, 3) (1, 3)} ——(1)
Take the RHS
(A × B) = {(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)}
(A × C) = {(0, 3)(0, 5) (1, 3) (1, 5)}
(A × B) n (A × C) = {(0, 3) (1, 3)} ——(2)
(1) = (2)
(iii) (A U B) × C = (A × C) U (B × C)
A = {0, 1} , B = {2, 3, 4} and C = {3, 5} .
A U B = {0, 1, 2, 3, 4}
(A U B) × C = { (0, 3)(0, 5)(1, 3)(1, 5)(2, 3)(2, 5)(3, 3)(3, 5)(4, 3)(4, 5) } —–(1)
(A × B) = {(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)}
(B × C) = {(2, 3) (2, 5) (3, 3)(3, 5) (4, 3) (4, 5)}
(A × C) U (B × C) = { (0, 3)(0, 5)(1, 3)(1, 5)(2, 3)(2, 5)(3, 3)(3, 5)(4, 3)(4, 5) } —–(2)
(1) = (2)
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Q. 7. Let A = The set of all natural numbers less than 8, B = The set of all Prime numbers less than 8, C = The set of even prime number. Verify that
