Define 'subset' as part of the definition and explain how subsets can be smaller than the original set. Provide examples.
Understand the Problem
The content appears to involve various mathematical concepts, such as subsets, elements, and definitions related to set theory or similar topics. It discusses how subsets are formed and how they relate to original sets, indicating that there may be examples or problems related to these concepts.
Answer
The number of subsets is $8$, and the proper subsets are $\{\}, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}$.
Answer for screen readers
The number of subsets of the set $A = {1, 2, 3}$ is $2^3 = 8$. The proper subsets are ${}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}$.
Steps to Solve
- Identify the main concepts of subsets
A subset is defined as a set where all of its elements are also contained within another set, known as the original set. For example, if $A = {1, 2, 3}$, then the subsets of $A$ include ${}$ (the empty set), ${1}$, ${2}$, ${3}$, ${1, 2}$, ${1, 3}$, ${2, 3}$, and ${1, 2, 3}$.
- Determine the number of subsets
The formula to calculate the number of subsets of a set with $n$ elements is given by the equation:
$$ \text{Number of subsets} = 2^n $$
Here, $n$ is the number of elements in the original set.
- Example calculation
For the set $A = {1, 2, 3}$, which has 3 elements, the number of subsets can be calculated as:
$$ \text{Number of subsets} = 2^3 = 8 $$
- Understanding proper subsets
A proper subset is a subset that contains at least one element but not all elements of the original set. For the set $A$, the proper subsets would be:
$$ {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} $$
Note that ${1, 2, 3}$ is not included as it is the entire set.
The number of subsets of the set $A = {1, 2, 3}$ is $2^3 = 8$. The proper subsets are ${}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}$.
More Information
This indicates that every set has a collection of subsets, including a single empty set and the set itself. Understanding subsets and proper subsets helps clarify foundational concepts in set theory.
Tips
- Mixing up the definitions of subsets and proper subsets. Remember that a proper subset cannot be equal to the original set.
- Forgetting to include the empty set as a subset.
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