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

Step 1: Understanding the Concept:
The problem asks for the number of subsets \(C\) of \(A\) that have at least one element in common with \(B\). It is easier to use the complementary counting principle: Total subsets minus subsets that have no elements in common with \(B\).
Step 3: Detailed Explanation:
1. Total number of subsets of \(A\):
\(A\) has 7 elements, so total subsets \(= 2^{7} = 128\).
2. Find elements in \(A\) that are NOT in \(B\):
\(B = \{3, 6, 7, 9\}\). The common elements are \(A \cap B = \{3, 6, 7\}\).
The elements of \(A\) not in \(B\) are \(A \setminus B = \{1, 2, 4, 5\}\).
3. Subsets of \(A\) with no common elements with \(B\):
These are subsets formed using only the elements in \(\{1, 2, 4, 5\}\).
Number of such subsets \(= 2^{4} = 16\).
4. Required number of subsets:
\[ \text{Result} = \text{Total Subsets} - \text{Subsets with null intersection} \]
\[ \text{Result} = 128 - 16 = 112 \]
Step 4: Final Answer:
The number of such subsets is 112.