Fewpal
+1 vote
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in Mathematics by (2.5k points)

Let A and B be sets. 

If A ∩ X = B ∩ X = Φ 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) 

1 Answer

+1 vote
by (2.5k points)
 
Best answer

To show: 

A = B It can be seen that 

A = A ∩ (A ∪ X) = A ∩ (B ∪ X)

 [A ∪ X = B ∪ X] = (A ∩ B) ∪ (A ∩ X)

[Distributive law] = (A ∩ B) ∪ Φ [A ∩ X = Φ] = A ∩ B …………………………………………………………….. (1) 

Now, 

B = B ∩ (B ∪ X) = B ∩ (A ∪ X) [A ∪ X = B ∪ X] = (B ∩ A) ∪ (B ∩ X) 

[Distributive law] = (B ∩ A) ∪ Φ [B ∩ X = Φ]  10 = B ∩ A = A ∩ B …………………………………………………………… (2) Hence, 

from (1) and (2), we obtain A = B. 

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