Question

Let \(A = \{1,2,3,4,5,6,7\}\) and \(B = \{3,6,7,9\}\). Then, the number of elements in the set \(\{ C \subseteq A : C \cap B \neq \phi \}\) is:

• A. 112
• B. 120
• C. 108
• D. 96

Hint:

Use complement: total subsets - subsets with no intersection.

Answer 1

The Correct Option is A

Concept: Total subsets: \(2^n\)

Step 1: \[ A = \{1,2,3,4,5,6,7\},\quad B \cap A = \{3,6,7\} \]

Step 2: Subsets with \(C \cap B = \phi\) means no element from \(\{3,6,7\}\) Remaining elements: \[ \{1,2,4,5\} \Rightarrow 4 \text{ elements} \] \[ \text{Number of such subsets} = 2^4 = 16 \]

Step 3: Total subsets of A: \[ 2^7 = 128 \]

Step 4: Required subsets: \[ 128 - 16 = 112 \]