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Chapter 1: Problem 93

Discussion. What is wrong with each statement? Explain. a) \(3 \subseteq\\{1,2,3\\}\) b) \(\\{3\\} \in\\{1,2,3\\}\) c) \(\varnothing=\\{\varnothing\\}\)

Short Answer

Expert verified
a) Use \( \in \) instead of \( \subseteq \). b) Use \( 3 \in \) instead of \( \{3\} \). c) \( \varnothing eq \{\varnothing\} \).

Step by step solution

01

Analyze statement (a)

Given statement: \(3 \subseteq\{1,2,3\}\)ewline The symbol \(\subseteq\) indicates a subset relationship. In set theory, an element is indicated using \(\in\) and a subset is a set itself. The correct statement should be \(\{3\} \subseteq\{1,2,3\}\). Thus, \(3\) is an element, not a set.
02

Analyze statement (b)

Given statement: \(\{3\} \in\{1,2,3\}\)ewline The symbol \(\in\) indicates that an element belongs to a set. Here, \{3\} represents a set containing the element \(3\). However, we need the element \(3\) itself, not a set that contains \(3\). The correct statement should be \(3 \in\{1,2,3\}\).
03

Analyze statement (c)

Given statement: \(\varnothing =\{\varnothing\}\)ewline The symbol \(=\) indicates that two sets are equal. In set theory, \( \{\textbackslash emptyset\} \) is a set containing an empty set as an element, whereas \(\emptyset\) is just the empty set with no elements. Therefore, an empty set is not equal to a set containing an empty set. They are fundamentally different sets.

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Key Concepts

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

Subset
Understanding the concept of a subset is crucial for mastering set theory. A set \(A\) is called a subset of another set \(B\) if every element of \(A\) is also an element of \(B\). This relationship is denoted by the symbol \( \subseteq \). For example, if \(A = \{1, 2\}\) and \(B = \{1, 2, 3\}\), then \(A \subseteq B \), because both 1 and 2 are in \(B\). However, it is incorrect to use an element like 3 on its own when talking about subsets. The corrected statement from the exercise should be \( \{3\} \subseteq \{1, 2, 3\} \) because \( \{3\} \) is a set, not just an element.
Element
Elements are the basic building blocks of sets. An element can be a number, a person, an object, or even another set. To indicate that an object is an element of a set, set theory uses the symbol \( \in \). For instance, if we have the set \( S = \{1, 2, 3\} \), we say that 2 \( \in \) \( S \) to show that 2 is one of the elements in the set \( S \). In contrast, a set containing that element, like \( \{2\} \), is not the same as 2 itself. Therefore, the exercise correction was necessary to show that \( 3 \in \{1, 2, 3\} \) is the accurate notation.
Empty Set
The empty set, also known as the null set, is a unique set with no elements. It is denoted by the symbol \( \varnothing \) or \( \{\} \). Because it contains no elements, it is a conceptually simple but pivotal object in set theory. It's crucial to distinguish between the empty set and a set containing the empty set as its only element. The notation \( \{\varnothing \} \) represents a set whose only element is the empty set, whereas \( \varnothing \) is just empty. The exercise correctly identified the mistake: \( \varnothing eq \{\varnothing \} \). While \( \varnothing \) has no elements, \( \{\varnothing \} \) has exactly one element, which is \( \varnothing \).

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Most popular questions from this chapter

Simplify each expression. $$ 5 w-2+4(w-3)-6(w-1) $$

Simplify each expression. $$ 6+5\left(k^{3}-2\right)-k^{3}+5 $$

Solve each problem. Saving for college. The average cost of a B.A. at a private college in 2021 will be \(\$ 100,000\) (U.S. Department of Education, www.ed.gov). The principal that must be invested at interest rate \(r\) compounded annually to have \(A\) dollars in \(n\) years is given by the algebraic expression $$ \frac{A}{(1+r)^{n}} $$ What amount must Melanie's generous grandfather invest in 2005 at \(7 \%\) compounded annually so that Melanie will have \(\$ 100,000\) for her college education in \(2021 ?\)

Use an operation with signed numbers to solve each problem and identify the operation used. The Jones family has a house that is worth \(\$ 85,000,\) but they still owe \(\$ 45,000\) on the mortgage. They have \(\$ 2300\) in credit card debt, \(\$ 1500\) in other debts, \(\$ 1200\) in savings, and two cars worth \(\$ 3500\) each. What is the net worth of the Jones family?

Perform these computations. $$-0.241-0.3$$
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