If F = {1, 2, 3, 4}, how many subsets of F contain the element 1?

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Question: If F = {1, 2, 3, 4}, how many subsets of F contain the element 1?

Options:

Correct Answer: 6

Solution:

If 1 is included, we can choose from the remaining elements {2, 3, 4}, which has 2^3 = 8 subsets.

If F = {1, 2, 3, 4}, how many subsets of F contain the element 1?

If F = {1, 2, 3, 4}, how many subsets of F contain the element 1?

Step 1: Identify the set F, which is {1, 2, 3, 4}.
Step 2: We want to find subsets of F that include the element 1.
Step 3: Since 1 must be included in every subset, we can focus on the remaining elements, which are {2, 3, 4}.
Step 4: Count the number of elements left after including 1. There are 3 elements: 2, 3, and 4.
Step 5: Each of the remaining elements can either be included in a subset or not. This gives us 2 choices (include or exclude) for each of the 3 elements.
Step 6: Calculate the total number of subsets that can be formed from the remaining elements. This is done using the formula 2^n, where n is the number of remaining elements. Here, n = 3, so we calculate 2^3.
Step 7: Calculate 2^3, which equals 8. This means there are 8 different subsets that can be formed with the elements {2, 3, 4}.
Step 8: Since all these subsets include the element 1, the total number of subsets of F that contain the element 1 is 8.

Subsets – Understanding how to calculate the number of subsets of a set, particularly when certain elements must be included.
Combinatorics – Applying the principles of combinatorics to determine the number of ways to select elements from a set.