If A = {{1, 2, 3}}, then how many elements are there in the power set of A?
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Understanding the Power Set of A = {{1, 2, 3}}
The question asks us to find the number of elements in the power set of a given set A, where A is defined as {{1, 2, 3}}. To solve this, we first need to understand what a power set is and how to determine the number of elements it contains.
What is a Power Set?
The power set of a set S is the set of all possible subsets of S, including the empty set and the set S itself. It is denoted by $P(S)$ or $2^S$.
Determining the Number of Elements in Set A
The given set is A = {{1, 2, 3}}. It is crucial to notice the structure of set A. Set A does not contain the numbers 1, 2, and 3 individually as its elements. Instead, set A contains only one element, which is the set {1, 2, 3}.
Let's denote the set {1, 2, 3} by B. So, A = {B}.
The number of elements in set A, also known as the cardinality of A and denoted by $n(A)$, is the count of distinct elements within the curly braces that define A. In this case, A contains only one element, the set {1, 2, 3}.
Therefore, $n(A) = 1$.
Calculating the Number of Elements in the Power Set of A
The number of elements in the power set of a set S with $n(S)$ elements is given by the formula $2^{n(S)}$.
For the given set A, we found that $n(A) = 1$.
Using the formula, the number of elements in the power set of A, $P(A)$, is:
Number of elements in $P(A) = 2^{n(A)}$
Number of elements in $P(A) = 2^1$
Number of elements in $P(A) = 2$
Listing the Elements of the Power Set of A
To confirm the result, let's list the elements of $P(A)$. The elements of $P(A)$ are all the subsets of A = {{1, 2, 3}}. The subsets are:
- The empty set ($\emptyset$). The empty set is a subset of every set.
- The set containing the single element of A, which is {{1, 2, 3}}.
So, the power set of A is $P(A) = \{\emptyset, \{\{1, 2, 3\}\}\}$.
Counting the elements in $P(A)$, we find there are exactly 2 elements.
Conclusion
The set A = {{1, 2, 3}} has only one element, which is the set {1, 2, 3}. The number of elements in its power set is $2^{n(A)} = 2^1 = 2$.
| Set | Elements | Cardinality ($n$) | Power Set ($P$) | Number of Elements in Power Set ($2^n$) |
|---|---|---|---|---|
| {{1, 2, 3}} | {{1, 2, 3}} (one element: the set {1, 2, 3}) | 1 | {$ \emptyset $, {{1, 2, 3}}} | $2^1 = 2$ |
Revision Table: Power Set Calculation
| Concept | Definition | Example for A = {{1, 2, 3}} |
|---|---|---|
| Set A | The given set | {{1, 2, 3}} |
| Cardinality of A ($n(A)$) | Number of elements in A | 1 (the element is the set {1, 2, 3}) |
| Power Set of A ($P(A)$) | Set of all subsets of A | {$ \emptyset $, {{1, 2, 3}}} |
| Number of elements in $P(A)$ | $2^{n(A)}$ | $2^1 = 2$ |
Additional Information on Sets and Power Sets
Understanding the difference between an element and a set containing that element is key in set theory. Here are some related concepts:
- Set: A well-defined collection of distinct objects. Objects in a set are called elements.
- Element: An object belonging to a set. For example, 1 is an element of {1, 2, 3}.
- Cardinality: The number of elements in a set. The cardinality of {a, b, c} is 3. The cardinality of {{a, b}} is 1.
- Subset: Set B is a subset of set A if all elements of B are also elements of A. The empty set ($\emptyset$) is a subset of every set. Every set is a subset of itself.
- Power Set: The set of all subsets of a given set. If a set has $n$ elements, its power set has $2^n$ elements.
- Set containing a set: A set can have other sets as its elements. In A = {{1, 2, 3}}, the element is the set {1, 2, 3}. This is different from the set {1, 2, 3}, whose elements are 1, 2, and 3.
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