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Chapter 2: Problem 49

In Exercises 41-54, determine whether each statement is true or false. If the statement is false, explain why. \(\\{5\\} \in\\{\\{5\\},\\{9\\}\\}\)

Short Answer

Expert verified
The statement is true, as \({5}\) is a member of the set \({\{5\}, \{9\}\}\)

Step by step solution

01

Understand Set Notation

Sets are a collection of items. They are generally grouped within braces '{}'. A set containing sets is also possible and the set \({\{5\}, \{9\}\}\) is a perfect example, where it contains two sets, \({5}\) and \({9}\).
02

Analyzing the Statement

We are to determine whether \({5}\) is a member of \({\{5\}, \{9\}\}\). In other words, we are checking if the set \({5}\) is present in the set \({\{5\}, \{9\}\}\). For a set to be a member of another set, it should be one of its elements.
03

Conclusion

The set \({5}\) is indeed a member of the set \({\{5\}, \{9\}\}\) because it is one of its elements. Therefore, the statement is true.

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Key Concepts

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

Understanding Set Notation
Set notation is a crucial part of set theory. Sets are collections of distinct objects, grouped within curly braces, such as \(\{a, b, c\}\). Each object within a set is called an "element" or "member." Sets can include numbers, letters, or even other sets. This notation is a way to describe the specific elements and their relationships.

There are different types of sets:
  • Finite sets: Sets with a limited number of elements, like \(\{1, 2, 3\}\).
  • Infinite sets: Sets with an unlimited number of elements, like \(\text{all natural numbers}\).
  • Empty set: A set with no elements, denoted as \(\emptyset\) or \(\{\}\).
Set notation helps in clarifying the structure and contents of a set, making it easier to analyze and understand. It's a language to communicate mathematical ideas, especially in problems involving collections like the exercise provided.
Defining a Member of a Set
A "member" or "element" of a set is any individual object contained within a set. If an object \(x\) is an element of a set \(A\), we write this as \(x \in A\). This means \(x\) is a part of \(A\).

In the context of our exercise, consider the set \(\{5, 9\}\):
  • 5 is a member of this set, which we write as \(5 \in \{5, 9\}\).
  • 3 is not a member, so we write \(3 otin \{5, 9\}\).
Members are the key components of sets. Identifying whether an object is a member involves checking its presence within the curly braces representing the set. Understanding this concept is essential for solving problems in set theory.
Applying Mathematical Reasoning
Mathematical reasoning is the process of thinking logically about problems to reach valid conclusions. When working with sets, reasoning helps ascertain relationships between members and sets themselves.

Consider the statement from the exercise: \(\{5\} \in \{\{5\}, \{9\}\}\):
  • We reason that \(\{5\}\) is an element of \(\{\{5\}, \{9\}\}\) because \(\{5\}\) is listed as part of the set.
  • This leads us to conclude that the statement is true.
Reasoning in set theory involves examining the relationships between elements and sets. Developing these skills allows you to approach problems more logically and confidently.
Understanding an Element of a Set
An "element" or "member" refers to any object within a set. Recognizing elements is key to set theory. Sets can contain numbers, letters, or even other sets as their elements.

In the exercise, we have a set \(\{\{5\}, \{9\}\}\):
  • The elements are \(\{5\}\) and \(\{9\}\), both of which are themselves sets.
  • The set \(\{5\}\) is an element, not the number 5 alone.
Understanding what constitutes an element is vital for correctly interpreting and solving problems. In set theory, precision in recognizing and defining elements ensures accuracy in reasoning and conclusions.

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Most popular questions from this chapter

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