Q. 114.2( 49 Votes )

# Let A and B be sets. If A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X, show that A = B.

(Hints A = A ∩ (A ∪ X) , B = B ∩ (B ∪ X) and use Distributive law)

Answer :

It is given in the question that,

A and B are sets such that

And, for some set X

Now, we have to show that A = B

As it can be seen that,

= (By using distributive law)

=

= (i)

Now, we have:

=

= (By using distributive law)

=

=

(ii)

∴ From the equation (i) and (ii), we have

A = B

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Let A = {a, b, c, d}, B = {c, d, e} and C = {d, e, f, g}. Then verify each of the following identities:

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × (B – C) = (A × B) – (A × C)

(iii) (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B)

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(i) n(A × B)

(ii) n(B × A)

(iii) n(A × B) ∩ (B × A)

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RS Aggarwal - Mathematics(i) If A ⊆ B, prove that A × C ⊆ B × C for any set C.

(ii) If A ⊆ B and C ⊆ D then prove that A × C ⊆ B × D.

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Using properties of sets prove the statements given

For all sets A and B, (A ∪ B) – B = A – B

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Hint n(A) = n(A – B) + n(A ∩ B) n(A ∩ B) = (23 – 8) = 15.

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