If `A=(1,2,3,4)`, then how many subsets of A contain the element 1 but not 4?

Options

A. 16
B. 4
C. 8
D. 24

Explanation

To find the number of subsets of the set \( A = \{1, 2, 3, 4\} \) that contain the element 1 but do not contain the element 4, we can follow these steps:

### Step 1: Identify the elements to consider
Since we want subsets that must include the element 1 and must exclude the element 4, we can focus on the remaining elements of the set \( A \) which are \( 2 \) and \( 3 \).

### Step 2: Create a new set
We can create a new set \( B = \{2, 3\} \) which contains the elements we can choose from, since we are already including 1 in every subset.

### Step 3: Determine the subsets of set B
The number of subsets of a set with \( n \) elements is given by \( 2^n \). Here, set \( B \) has 2 elements (2 and 3).
Thus, the number of subsets of \( B \) is:
\[ 2^2 = 4 \]

### Step 4: Include the element 1 in each subset
Each of these subsets of \( B \) can be combined with the element 1. Therefore, we can list the subsets that include 1:
1. \( \{1\} \)
2. \( \{1, 2\} \)
3. \( \{1, 3\} \)
4. \( \{1, 2, 3\} \)

### Step 5: Conclusion
Thus, the total number of subsets of \( A \) that contain the element 1 but do not contain the element 4 is \( 4 \).

### Final Answer
The number of subsets of \( A \) that contain the element 1 but not 4 is **4**.

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