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Sets Practice Test 1

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Question : 7 of 33

Marks: +1, -0

If A = {1, 3, 5, 7, 11, 13, 15, 17}, B = {2, 4, 6,8, 9, 10, 12, 14, 16, 8} and N is a UniversalSet, then (A′ ∩ ((A ∪ B) ∩ B) is:

A
N
B
ϕ
None of these

Solution: 👈: Video Solution

Concept:

Use set identities: when

A

and

B

are disjoint,

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

, and

A \cap A' = \varnothing

.

Explanation:

Given

A = \{1,3,5,7,11,13,15,17\}

.

B = \{2,4,6,8,9,10,12,14,16\}

(duplicate

8

is ignored).

A \cap B = \varnothing

, so

A

and

B

are disjoint sets.

Simplify

(A \cup B) \cap B'

using distributive law:

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

.

Since

B \cap B' = \varnothing

, this becomes

A \cap B'

.

As

A \cap B = \varnothing

, we have

A \subseteq B'

, so

A \cap B' = A

.

Thus

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

.

Now evaluate

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

.

By complement law,

A' \cap A = \varnothing

.

Hence the result is

\phi

.

Answer:

\phi

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