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Chapter 2: Problem 59

List all the subsets of the given set. \(\\{0\\}\)

Short Answer

Expert verified
The subsets of the given set {0} are {}, {0}.

Step by step solution

01

Identify the Number of Subsets

The principle of subsets states that a set with 'n' elements has 2^n subsets. Here, our given set has 1 element, so it has \(2^1 = 2\) subsets.
02

List the Subsets

Following the principle of subset creation, let's list all the subsets. The two subsets of the given set {0} are: the empty set and the set itself i.e., {}, {0}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subsets
In set theory, a subset is a set whose elements are entirely contained within another set, which is called the super-set. Imagine a bag containing different items. A subset would be like taking some or all of those items out of the bag. If you start with a set that contains just one element, like \(\{0\}\), you can create its subsets by considering any possible groupings of its elements.

Every set is a subset of itself, including the empty set. So, for \(\{0\}\), the subsets are \(\{0\}\) and \(\{\}\), the empty set. Subsets are fundamental in understanding how different sets relate to each other.
Empty Set
The empty set, often denoted as \(\{\}\) or \(\emptyset\), is a special set containing no elements. It鈥檚 like having an empty box with nothing in it. Despite its emptiness, it's considered a subset of every set, including itself.

The reason it's a subset of every set is due to how subsets are defined. Since the empty set contains no elements, there's nothing to challenge its status as a subset. So when listing subsets of any set, the empty set always makes an appearance.
Principle of Subsets
The principle of subsets is a handy rule in mathematics that helps quickly determine how many subsets a given set can have. For a set with \(n\) elements, it will have \(2^n\) subsets. This principle uses the power of two because each element can either be included or excluded from the subset.

In our example with the set \(\{0\}\), since there's only one element, \(2^1\) gives us 2 subsets. These are \(\{\}\) and \(\{0\}\). This principle is essential for understanding combinations and different ways to organize or select items from a set.
Mathematical Concepts
When learning about sets, it's essential to dive into several mathematical concepts that surround them. From understanding basic set operations like union and intersection to more complex ideas such as Cartesian products, sets form a backbone for many areas of mathematics.

Key concepts include:
  • Elements: The distinct objects or members within a set.
  • Universal Set: The complete set that contains all objects under consideration, allowing for comparison.
  • Power Set: The set of all subsets of a set, including the empty set and the set itself.
By building foundational knowledge of these concepts, students can better grasp how sets relate to each other and apply this understanding in various mathematical fields.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(A \cap A^{\prime}=\varnothing\)

Describe what is meant by the union of two sets. Give an example.

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(B \cap U\)

If \(A \cap B=\varnothing\), then \(n(A \cup B)=n(A)+n(B)\).

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cup B^{\prime}\)
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