Consider the proper subsets of {1, 2, 3, 4}. How many of these proper subsets are a superset of the set {3}?
7
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
| Subset | Contains 3? (Superset of {3}?) | Proper Subset of {1,2,3,4}? | Meets Both Conditions? |
|---|---|---|---|
| {} | No | Yes | No |
| {1} | No | Yes | No |
| {2} | No | Yes | No |
| {3} | Yes | Yes | Yes |
| {4} | No | Yes | No |
| {1,2} | No | Yes | No |
| {1,3} | Yes | Yes | Yes |
| {1,4} | No | Yes | No |
| {2,3} | Yes | Yes | Yes |
| {2,4} | No | Yes | No |
| {3,4} | Yes | Yes | Yes |
| {1,2,3} | Yes | Yes | Yes |
| {1,2,4} | No | Yes | No |
| {1,3,4} | Yes | Yes | Yes |
| {2,3,4} | Yes | Yes | Yes |
| {1,2,3,4} | Yes | No | No |
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
| Concept | Definition | Example (for S={1,2,3}) |
|---|---|---|
| Set | A collection of distinct objects. | {1, 2, 3} |
| Subset (\(\subseteq\)) | A is a subset of B if every element in A is also in B. | {1, 2} \(\subseteq\) {1, 2, 3} |
| Proper Subset (\(\subset\)) | A is a subset of B, and A \(\neq\) B. | {1, 2} \(\subset\) {1, 2, 3} |
| Superset (\(\supseteq\)) | A is a superset of B if B is a subset of A. | {1, 2, 3} \(\supseteq\) {1, 2} |
| Cardinality (|\(A\)|) | The number of elements in set A. | |\({1, 2, 3}\)| = 3 |
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, \(8 - 1 = 7\).
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