Understanding Proper Subsets and Supersets

This question asks us to find the number of proper subsets of the set {1, 2, 3, 4} that are also a superset of the set {3}. Let's break down the terms:

• Subset: A set A is a subset of a set B if all elements of A are also elements of B.
• Proper Subset: A set A is a proper subset of a set B if A is a subset of B, and A is not equal to B. In other words, B must contain at least one element not in A.
• Superset: A set A is a superset of a set B if B is a subset of A. This means all elements of B are contained within A. In this question, we need subsets that are a superset of {3}, meaning the subset must contain the element 3.

Finding Subsets of {1, 2, 3, 4}

The given set is S = {1, 2, 3, 4}. The number of elements in this set is n = 4.

The total number of subsets of a set with n elements is given by \(2^n\).

For S = {1, 2, 3, 4}, the total number of subsets is \(2^4 = 16\).

Let's list all the subsets:

• {} (the empty set)
• {1}, {2}, {3}, {4} (subsets with one element)
• {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} (subsets with two elements)
• {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4} (subsets with three elements)
• {1,2,3,4} (subset with four elements, which is the set itself)

Identifying Proper Subsets

A proper subset is any subset except the set itself. So, we exclude the set {1, 2, 3, 4} from the list of subsets.

The number of proper subsets is \(2^n - 1 = 2^4 - 1 = 16 - 1 = 15\).

The proper subsets are:

• {}
• {1}, {2}, {3}, {4}
• {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}
• {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}

Finding Proper Subsets that are a Superset of {3}

We need to find the proper subsets from the list above that contain the element 3. These are the subsets that are a superset of {3}.

Let's go through the list of proper subsets and check which ones contain 3:

• {} - Does not contain 3
• {1} - Does not contain 3
• {2} - Does not contain 3
• {3} - Contains 3
• {4} - Does not contain 3
• {1,2} - Does not contain 3
• {1,3} - Contains 3
• {1,4} - Does not contain 3
• {2,3} - Contains 3
• {2,4} - Does not contain 3
• {3,4} - Contains 3
• {1,2,3} - Contains 3
• {1,2,4} - Does not contain 3
• {1,3,4} - Contains 3
• {2,3,4} - Contains 3

The proper subsets that are also a superset of {3} are: {3}, {1,3}, {2,3}, {3,4}, {1,2,3}, {1,3,4}, {2,3,4}.

Counting the Subsets

Let's count the identified subsets:

• {3}
• {1,3}
• {2,3}
• {3,4}
• {1,2,3}
• {1,3,4}
• {2,3,4}

There are 7 such subsets.

Alternatively, consider subsets of {1, 2, 3, 4} that contain 3. To form such a subset, 3 must be included. The remaining elements {1, 2, 4} can either be included or excluded. The number of ways to choose elements from {1, 2, 4} is \(2^3 = 8\).

These 8 subsets containing 3 are:

• {3} (taking none from {1, 2, 4})
• {1,3}, {2,3}, {4,3} (taking one from {1, 2, 4})
• {1,2,3}, {1,4,3}, {2,4,3} (taking two from {1, 2, 4})
• {1,2,4,3} (taking all three from {1, 2, 4})

These are {3}, {1,3}, {2,3}, {4,3} (={3,4}), {1,2,3}, {1,4,3} (={1,3,4}), {2,4,3} (={2,3,4}), {1,2,4,3} (={1,2,3,4}).

These are the 8 subsets of {1, 2, 3, 4} that contain 3 (i.e., are a superset of {3}). We need the ones that are also proper subsets. This means we exclude the set {1, 2, 3, 4} itself.

So, the proper subsets that contain 3 are the 8 subsets that contain 3, minus the set {1, 2, 3, 4}. This gives \(8 - 1 = 7\) subsets.

Summary Table

Counting the rows where "Meets Both Conditions?" is "Yes", we find there are 7 such subsets.

Conclusion

There are 7 proper subsets of {1, 2, 3, 4} that are also a superset of the set {3}.

Revision Table: Set Theory Concepts

Additional Information: Counting Subsets with Specific Elements

To find the number of subsets of a set S with n elements that contain a specific element 'x':

Consider the elements of S other than 'x'. There are \(n-1\) such elements. Any subset containing 'x' is formed by taking 'x' and then choosing any combination of the remaining \(n-1\) elements.

The number of subsets of the remaining \(n-1\) elements is \(2^{n-1}\). Each of these combinations, when combined with 'x', forms a unique subset of S that contains 'x'.

Therefore, the number of subsets of a set with n elements that contain a specific element is \(2^{n-1}\).

In our problem, the set is {1, 2, 3, 4} (n=4) and the specific element is 3. The number of subsets containing 3 is \(2^{4-1} = 2^3 = 8\).

These 8 subsets are the supersets of {3}. We needed the ones that are also proper subsets, which means we exclude the original set {1, 2, 3, 4}. So, \(8 - 1 = 7\).