Let A and B be sets. If A∩X=B∩X=φand A∪X=B∪Xfor some set X, show that A=B.(Hints A=A∩(A∪X),B=B∩(B∪X)and use Distributive law)

Solution: A=A∩(A∪X) =A∩(B∪X) =(A∩B)∪(A∩X) =(A∩B)∪ϕ (by distributive law) A=A∩B eqn(1)now B=B∩(B∪X) =B∩(A∪X) =(B∩A)∪(B∩X) =(B∩A)∪ϕ B=(B∩A) eqn(2)so by eqn 1 & 2, A=B hence proved

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Let A and B be sets. If {A}{X}={B}{X}= {A}{X}={B}{X}for some s

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Let A and B be sets. If $A \cap X = B \cap X = φ$ and $A \cup X = B \cup X$ for some set X, show that $A = B$ .(Hints $A = A \cap (A \cup X), B = B \cap (B \cup X)$ and use Distributive law)

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Michael Discussed

Let A and B be sets. If

A \cap X = B \cap X = φ

and

A \cup X = B \cup X

for some set X, show that

A = B

.(Hints

A = A \cap (A \cup X), B = B \cap (B \cup X)

and use Distributive law)

7 mins ago

Discuss this question LIVE

7 mins ago

Text solutionVerified

Solution:

A = A \cap (A \cup X)

= A \cap (B \cup X)

= (A \cap B) \cup (A \cap X)

= (A \cap B) \cup ϕ

(by distributive law)

A = A \cap B

eqn(1)
now

B = B \cap (B \cup X)

= B \cap (A \cup X)

= (B \cap A) \cup (B \cap X)

= (B \cap A) \cup ϕ

= (B \cap A)

eqn(2)
so by eqn 1 & 2,

A = B

hence proved

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Question Text	Let A and B be sets. If $A \cap X = B \cap X = φ$ and $A \cup X = B \cup X$ for some set X, show that $A = B$ .(Hints $A = A \cap (A \cup X), B = B \cap (B \cup X)$ and use Distributive law)
Answer Type	Text solution:1
Upvotes	150

Let A and B be sets. If A∩X=B∩X=φand A∪X=B∪Xfor some set X, show that A=B.(Hints A=A∩(A∪X),B=B∩(B∪X)and use Distributive law)

Text solutionVerified

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Let A and B be sets. If $A \cap X = B \cap X = φ$ and $A \cup X = B \cup X$ for some set X, show that $A = B$ .(Hints $A = A \cap (A \cup X), B = B \cap (B \cup X)$ and use Distributive law)