/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Mark each as true or false. $$... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 21

Mark each as true or false. $$ \emptyset \in\\{\varnothing\\} $$

Short Answer

Expert verified
The statement \(\emptyset \in \{\emptyset\}\) is true, as the set containing the empty set \(\{\emptyset\}\) has one element, which is the empty set itself.

Step by step solution

01

Understanding the empty set and the set containing the empty set

The empty set, denoted by \(\emptyset\), is a set with no elements in it. On the other hand, the set containing the empty set, \(\{\emptyset\}\), has one element, which is the empty set itself.
02

Comparing the two sets

Now let's determine if the given statement, \(\emptyset \in \{\emptyset\}\), is true or false. The statement is saying that the empty set is an element of the set containing the empty set. Recall that the set containing the empty set, \(\{\emptyset\}\), has one element, which is the empty set itself, i.e., \(\{\emptyset\} = \{ \emptyset \} \). As we can see, the empty set is indeed an element of the set containing the empty set.
03

Conclusion

Therefore, the statement \(\emptyset \in \{\emptyset\}\) is true.

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Key Concepts

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

Set Theory
Set theory is a fundamental part of mathematics that deals with the collection of objects, known as elements or members. In study or application, sets are often denoted with curly braces, such as \( \{a, b, c\} \), and each element is listed inside the braces, separated by commas.

One of the most basic concepts in set theory is the empty set, symbolized by \( \emptyset \). It represents a set with no elements. It's important to distinguish the empty set from a set that contains the empty set, as the latter still has an element鈥擻( \emptyset \) itself鈥攖hough this element denotes 'nothingness'. Understanding these foundational concepts is crucial for further mathematical explorations, including functions, probability, and algebra.
Elements of a Set
In set theory, an element is a singular object contained within a set. The notation \( a \in A \) is used to denote that 'a' is an element of the set 'A'. Conversely, if 'a' is not in 'A', we write \( a otin A \).

An element can be anything: a number, a letter, a symbol, or even another set. This leads to the possibility of sets containing sets, known as nested sets. While the elements in a set are distinct, meaning no two elements can be the same, a set can have any number of elements, from zero (the empty set) to an infinite number. Understanding the unique identity of each element is essential for grasping more complex concepts like subsets, unions, intersections, and Cartesian products in set theory.
Subsets
A subset is a set whose elements are all contained within another set. If we have two sets, 'A' and 'B', we say that 'A' is a subset of 'B', denoted by \( A \subseteq B \), if every element of 'A' is also in 'B'. It's essential to note that any set is a subset of itself, and the empty set, being devoid of elements, is a subset of every set.

Special Consideration for the Empty Set

Considering the unique nature of the empty set, it may seem counterintuitive, but it is indeed a subset of any set because there are no elements in the empty set that could possibly not be in any other set. This fact reinforces the idea that even though the empty set contains nothing, it plays an integral role in the structure of set theory.

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Most popular questions from this chapter

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Three

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The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Five
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