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Sets and Relations question

2022 · 26 Jul · Shift 2 · Q34

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JEE Main
Mathematics
Sets and Relations
2022 · 26 Jul · Shift 2 · Q34

Sets and Relations question

2022 · 26 Jul · Shift 2 · Q34

JEE MainMathematicsSets and RelationsNumerical+4 / −1

Let

A=\{1,2,3,4,5,6,7\}

and

B=\{3,6,7,9\}

. Then the number of elements in the set

\{C \subseteq A: C \cap B eq \phi\}

is

\underline{\hspace{2cm}}

.

Numerical answer

View written solutionFree

Correct answer: 112

We need to count all subsets $C \subseteq A$ such that $C \cap B \neq \varnothing.$
First identify the elements of $B$ that are actually relevant for subsets of $A$ .

Given: $A=\{1,2,3,4,5,6,7\}, \qquad B=\{3,6,7,9\}.$

Since $C \subseteq A$ , the intersection $C \cap B$ can only involve elements common to both $A$ and $B$ . So, $A \cap B = \{3,6,7\}.$

Thus the condition $C \cap B \neq \varnothing$ means:

subset $C$ must contain at least one of $3,6,7$ .

Total number of subsets of $A$ : $2^7 = 128.$
Count subsets of $A$ that contain none of $3,6,7$ .

If $C$ contains none of these three elements, then $C$ can only be formed from $A \setminus \{3,6,7\} = \{1,2,4,5\}.$

Number of such subsets: $2^4 = 16.$

Therefore, required number of subsets is $128 - 16 = 112.$

So the number of elements in $\{C \subseteq A : C \cap B \neq \varnothing\}$ is $\boxed{112}.$

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