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X∩A=Y∩A=φ and X∪A=Y∪A Imply X=Y | JEE Sets

JEE Maths question with a full step-by-step solution.

Question

Let

X

and

Y

be two non-empty sets such that

X \cap A = Y \cap A = \phi

and

X \cup A = Y \cup A

for some non-empty set

A

. Then:

X

is a proper subset of

Y

Y

is a proper subset of

X

X = Y

correct

X

and

Y

are disjoint sets

Solution

X \cap A = Y \cap A = \phi \Rightarrow X, Y

are each disjoint from

A

. In

X \cup A = Y \cup A

, the common part

A

overlaps neither

X

nor

Y

, so removing

A

from both sides leaves

X = Y.

Correct answer: (3)

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