Question

Let \( A=\{2,4,6,8,10,12\} \), \( B=\{3,6,9,12\} \). How many subsets of B are not subsets of A?

• A. 12
• B. 11
• C. 10
• D. 13

Hint:

Using the complement method (Total - Unwanted) is often faster than counting directly. Here, counting subsets containing at least one element from \( B-A \) is harder than Total - (Subsets from \( A \cap B \)).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to count the number of subsets of B that are NOT completely contained within A. This is equivalent to: (Total subsets of B) - (Subsets of B that ARE subsets of A). Step 2: Detailed Calculation:
1. Total subsets of B: Set \( B \) has 4 elements: \(\{3, 6, 9, 12\}\). Total subsets = \( 2^4 = 16 \). 2. Subsets of B that are subsets of A: A subset of B is a subset of A only if all its elements belong to A. Thus, we look for elements in the intersection \( A \cap B \). \( A \cap B = \{6, 12\} \). The number of elements in the intersection is 2. The number of subsets formed using only these elements is \( 2^2 = 4 \). 3. Subsets NOT in A: \[ \text{Required Count} = \text{Total Subsets} - \text{Subset in A} \] \[ \text{Required Count} = 16 - 4 = 12 \] Step 4: Final Answer:
There are 12 such subsets.