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Simplifying (A - B) union (B - A) union (A complement intersect B complement) | JEE Advanced

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

Question

Let

A

and

B

be two sets. Then the set

(A-B) \cup (B-A) \cup \left(\overline{A}\cap\overline{B}\right)

is the same as

\overline{A}\cup\overline{B}

correct

\overline{\left(A\cup B\right)}

\phi

U

(here

U

denotes the universal set, and

A

B

are subsets of

U

)

Solution

Step 1:

A - B = A \cap \overline{B}, \qquad B - A = B \cap \overline{A}

Step 2: A point of

(A-B)\cup(B-A)

lies in exactly one of

A

and

B

, so

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

Step 3:

\overline{A}\cap\overline{B} = \overline{A\cup B}

by De Morgan's law, which is everything outside

A\cup B

. Step 4: The first two pieces give

A\cup B

without the overlap and the third gives everything outside

A\cup B

, so

\Big[(A\cup B)-(A\cap B)\Big] \cup \overline{A\cup B} = U - (A\cap B)

Step 5:

U - (A\cap B) = \overline{A\cap B} = \overline{A}\cup\overline{B}

again by De Morgan's law. Step 6:

\overline{A}\cup\overline{B}

is option (1). Option (2) is only the third piece

\overline{A\cup B}

by itself; option (3) would need

A\cap B = U

; option (4) would need

A\cap B

to be included, and none of the three pieces contains a point of

A\cap B

. Answer: (1).

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