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Chapter 2: Problem 50

In Exercises 41-54, determine whether each statement is true or false. If the statement is false, explain why. \(\\{1\\} \in\\{\\{1\\},\\{3\\}\\}\)

Short Answer

Expert verified
True. The set \(\{1\}\) is an element of the set \(\{\{1\},\{3\}\}\).

Step by step solution

01

Understand the Concept

The '\(\in\)' symbol is used to denote that an element belongs to a set. A set is a collection of distinct objects, which can also be sets by themselves. In this exercise, we're asked whether the set \(\{1\}\) is an element of the set \(\{\{1\},\{3\}\}\), not whether 1 is an element of the set.
02

Verify the Membership

Check to see if any elements in the larger set \(\{\{1\},\{3\}\}\) is identical to the set \(\{1\}\). Here, we can see that one of the elements in the larger set is exactly the set \(\{1\}\), so our statement \(\{1\} \in \{\{1\},\{3\}\}\) is true.

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Key Concepts

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

Element of a Set
In set theory, an "element" is simply a distinct object or item that forms part of a set. A set is like a collection of distinct items. These items can be anything you can think of, like numbers, letters, or even other sets. When we say something is an "element of a set," we mean that the object or item is included in that set. For example:
  • The number 2 is an element of the set \(\{1, 2, 3\}\).
  • The letter 'a' is an element of the set \(\{a, b, c\}\).
  • The set \(\{1\}\) itself can be an element of another set, like \(\{\{1\}, \{2\}\}\).
Understanding elements is key to grasping how sets work. Just remember that elements are the building blocks within sets. In the exercise provided, \(\{1\}\) is considered as a single element within another set.
Membership
Membership in set theory deals with determining whether an item belongs to a particular set. When we use the term membership, we're talking about identifying elements that are present within a set. The symbol \(\in\) is used to identify this relationship.
  • If we say \(a \in S\), it means that \(a\) is an element of set \(S\).
  • If \(a\) is not an element of set \(S\), we write \(a otin S\).
In this context, the membership described in the exercise involves checking if the set \(\{1\}\) is part of the larger set \(\{\{1\}, \{3\}\}\). As demonstrated, since \(\{1\}\) exactly matches one of the elements of the larger set, \(\{1\} \in \{\{1\}, \{3\}\}\) is indeed true. Knowing how to test membership answers questions like "Is this object part of that collection?" and is fundamental to set logic.
Mathematical Notation
Mathematical notation refers to a system of symbols used to denote sets, elements, and their relationships. It's like a language for math that allows us to express complex ideas succinctly and clarify their meaning. Here are some important symbols in set theory:
  • Curly braces \(\{\}\) denote a set. Anything between these braces is included in the set.
  • The symbol \(\in\) signifies that an element is a member of a set.
  • The symbol \(otin\) is used to indicate that an element is not a member of a set.
Using these symbols, we can succinctly describe relationships without lengthy explanations. For example, \(\{2, 3\} otin \{1, \{2, 3\}\}\) clearly confirms that the set \(\{2, 3\}\) is not an element of the set \(\{1, \{2, 3\}\}\). In the exercise, the use of notation helps to express the idea that the set \(\{1\}\) is a member of another set using \(\in\), making the solution precise and unambiguous.

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Most popular questions from this chapter

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