Consider all the subsets of the set A = {1, 2, 3, 4}. How many of them are supersets of the set {4}?
8
Understanding Subsets and Supersets
This question asks us to find the number of subsets of the set $A = \{1, 2, 3, 4\}$ that are also supersets of the set $\{4\}$. Let's break down the terms involved.
- Subset: A set B is a subset of a set A if every element in B is also in A. We denote this as $B \subseteq A$.
- Superset: A set A is a superset of a set B if every element in B is also in A. This is the same relationship as subset, just viewed from the other side. If $B \subseteq A$, then A is a superset of B. We denote this as $A \supseteq B$.
In this problem, we are looking for subsets of $A = \{1, 2, 3, 4\}$ that are supersets of the set $\{4\}$. This means we are looking for subsets $S$ of $A$ such that $\{4\} \subseteq S$. For $\{4\}$ to be a subset of $S$, the element $4$ must be present in $S$.
Identifying Required Subsets
We need to find all subsets of $A = \{1, 2, 3, 4\}$ that contain the element $4$. A subset of $A$ is formed by choosing some elements from $A$. For the subset to be a superset of $\{4\}$, the element $4$ must always be included.
The set $A$ has elements $\{1, 2, 3, 4\}$. If a subset of A must contain $4$, then we are free to choose whether to include the other elements from A, which are $1, 2,$ and $3$. The possible elements we can choose from, besides the mandatory $4$, are $\{1, 2, 3\}$.
Any subset of $\{1, 2, 3\}$ can be combined with the element $4$ to form a subset of $A$ that contains $4$. The number of subsets of a set with $n$ elements is $2^n$. The set $\{1, 2, 3\}$ has $3$ elements.
Number of subsets of $\{1, 2, 3\}$ is $2^{|\{1, 2, 3\}|} = 2^3 = 8$.
Constructing the Subsets
Let's list the subsets of $\{1, 2, 3\}$ and then form the corresponding supersets of $\{4\}$ that are subsets of $A$:
| Subsets of $\{1, 2, 3\}$ | Corresponding Subset of $A$ (Superset of $\{4\}$) |
|---|---|
| $\emptyset$ | $\emptyset \cup \{4\} = \{4\}$ |
| $\{1\}$ | $\{1\} \cup \{4\} = \{1, 4\}$ |
| $\{2\}$ | $\{2\} \cup \{4\} = \{2, 4\}$ |
| $\{3\}$ | $\{3\} \cup \{4\} = \{3, 4\}$ |
| $\{1, 2\}$ | $\{1, 2\} \cup \{4\} = \{1, 2, 4\}$ |
| $\{1, 3\}$ | $\{1, 3\} \cup \{4\} = \{1, 3, 4\}$ |
| $\{2, 3\}$ | $\{2, 3\} \cup \{4\} = \{2, 3, 4\}$ |
| $\{1, 2, 3\}$ | $\{1, 2, 3\} \cup \{4\} = \{1, 2, 3, 4\}$ |
As shown in the table, there are $8$ such subsets of $A$ that contain the element $4$. Each of these $8$ subsets is a superset of $\{4\}$.
Final Count
The number of subsets of the set $A = \{1, 2, 3, 4\}$ that are supersets of the set $\{4\}$ is equal to the number of subsets of the set $\{1, 2, 3\}$, which is $2^3 = 8$.
Therefore, there are 8 subsets of $A = \{1, 2, 3, 4\}$ that are supersets of $\{4\}$.
Revision Table: Key Concepts
| Term | Definition | Example (with $A=\{1,2,3,4\}, B=\{4\}$) |
|---|---|---|
| Subset ($\subseteq$) | Set X is a subset of Set Y if every element of X is in Y. | $\{4\} \subseteq \{1, 2, 3, 4\}$ (True) $\{5\} \subseteq \{1, 2, 3, 4\}$ (False) |
| Superset ($\supseteq$) | Set Y is a superset of Set X if every element of X is in Y. | $\{1, 2, 3, 4\} \supseteq \{4\}$ (True) $\{1, 2\} \supseteq \{4\}$ (False) |
| Power Set ($\mathcal{P}(A)$) | The set of all subsets of A. | $\mathcal{P}(\{4\}) = \{\emptyset, \{4\}\}$ $\mathcal{P}(\{1, 2\}) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$ |
Additional Information: Set Theory Basics
Set theory is a fundamental branch of mathematics that deals with collections of objects. Understanding concepts like sets, subsets, supersets, union, intersection, etc., is crucial for various mathematical and computer science topics.
- The number of subsets of a set with $n$ elements is $2^n$. This is because for each element, you have two choices: either include it in the subset or not.
- If you need to find the number of subsets of set A that contain a specific element 'x', you can think of it as forming subsets from the set A excluding 'x', and then adding 'x' to each of those subsets. The number of elements in A excluding 'x' is $|A| - 1$. So, the number of such subsets is $2^{|A|-1}$. In our problem, $|A|=4$ and the specific element is $4$. The number of subsets of $A$ containing $4$ is $2^{4-1} = 2^3 = 8$.
- The smallest subset of any non-empty set is the empty set, $\emptyset$. The largest subset is the set itself.
Important Questions from Operations on Sets
What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?
If A = {x ∈ R : x 2+ 6x - 7 < 0} and B = {x ∈ R : x 2+ 9x + 14 > 0}, then which of the following is/are correct?
1. (A ∩ B) = (-2, 1)
2. (A - B) = (-7, -2)
Select the correct answer using the code given below:A, B, C and D are four sets such that A ∩ B = C ∩ D = ϕ. Consider the following:
1. A ∪ C and B ∪ D are always disjoint.
2. A ∩ C and B ∩ D are always disjoint.
Which of the above statements is/are correct?A coin is tossed three times. Consider the following events:
A: No head appears
B: Exactly one head appears
C. At least two heads appear
Which one of the following is correct?
If C = { 2, 4, 6, 8, 10, 12, 14, 16 }, and D = {5, 10, 15, 20}, then the number of elements in the set D - C is:
