Question

26. Let A and B be two sets having p and q elements respectively such that q – p < 0.

Write the

(a) minimum number of elements that A U B can have.

(b) maximum number of elements that A U B can have.

(c) minimum number of elements that A ∩ B can have.

(d) maximum number of elements that A ∩ B can have.​

Answer 1

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GIVEN

Let A and B be two sets having p and q elements respectively such that q – p < 0

TO DETERMINE

(a) Minimum number of elements of A U B

(b) Maximum number of elements of A U B

(c) Minimum number of elements of A ∩ B

(d) Maximum number of elements of A ∩ B

CALCULATION

Since A and B be two sets having p and q elements respectively such that p > q

So n ( A ) = p , n( B ) = q such that p > q

Now in context of the given problem we have from Set theory that

$\sf{ \Phi \subset ( \: A \cap \: B \: ) \subset A} \: \: \: and$

$\sf{ \Phi \subset \: ( \: A \cap \: B \: ) \subset B}$

Hence

$\sf{n( \Phi ) \leqslant \:n ( \: A \cap \: B \: ) \leqslant min \{ n (A) \: ,\: n( B) \}}$

So

$\sf{0 \leqslant \:n ( \: A \cap \: B \: ) \leqslant min \{ n (A) \: ,\: n( B) \}} \: \: ...(1)$

Again

$\sf{n(A \cup \: B) = n( A) + n(B) - n( A \cap B)}$

So

$\sf{ \max \{ \: n( A) \: , \: n(B) \} \leqslant n(A \cup \: B) \leqslant n( A) + n(B) } \: \: .....(2)$

(a) From Equation (2)

Minimum number of elements of A U B = p

(b) From Equation (2)

Maximum number of elements of A U B

= p+ q

(c) From Equation (1)

Minimum number of elements of A ∩ B = 0

(d) From Equation (1)

Maximum number of elements of A ∩ B = q

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LEARN MORE FROM BRAINLY

If n(A) = 300, n(A∪B) = 500, n(A∩B) = 50 and n(B′) = 350, find n(B) and n(U)

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