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Questions and Answers
What does the notation x ∈ S represent?
What does the notation x ∈ S represent?
Which of the following represents a set containing only the integer 0?
Which of the following represents a set containing only the integer 0?
How many distinct elements are there in the set {a, {a, b}, {a}}?
How many distinct elements are there in the set {a, {a, b}, {a}}?
What does the expression Uₙ = n, -n represent for a nonnegative integer n?
What does the expression Uₙ = n, -n represent for a nonnegative integer n?
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How can infinite sets be denoted using set-roster notation?
How can infinite sets be denoted using set-roster notation?
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Which notation correctly represents the set of all integers?
Which notation correctly represents the set of all integers?
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In set-builder notation, how is a new set defined?
In set-builder notation, how is a new set defined?
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What is the relation between the sets A, B, and C if A = {1, 2, 3}, B = {2, 3, 1}, and C = {1, 1, 2, 3, 3, 3}?
What is the relation between the sets A, B, and C if A = {1, 2, 3}, B = {2, 3, 1}, and C = {1, 1, 2, 3, 3, 3}?
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Which of the following represents an open interval of real numbers between -5 and 1?
Which of the following represents an open interval of real numbers between -5 and 1?
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If A = {𝑥 ∈ 𝑍 | 3 ≤ 𝑥 < 8} and B = {𝑥 ∈ 𝑍 | 0 ≤ 𝑥 ≤ 7}, which statement is true?
If A = {𝑥 ∈ 𝑍 | 3 ≤ 𝑥 < 8} and B = {𝑥 ∈ 𝑍 | 0 ≤ 𝑥 ≤ 7}, which statement is true?
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Which of the following correctly describes the set {𝑥 ∈ 𝑍 | -1 ≤ 𝑥 < 6}?
Which of the following correctly describes the set {𝑥 ∈ 𝑍 | -1 ≤ 𝑥 < 6}?
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What does the notation A ⊆ B imply?
What does the notation A ⊆ B imply?
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Which of the following statements is false regarding the set C = {100, 200, 300, 400, 500}?
Which of the following statements is false regarding the set C = {100, 200, 300, 400, 500}?
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Which set represents integers that are not positive and are greater than or equal to -4 and less than or equal to 0?
Which set represents integers that are not positive and are greater than or equal to -4 and less than or equal to 0?
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If A = {2, 2, 2} and B = {2, 2, 2}, which of the following statements is true?
If A = {2, 2, 2} and B = {2, 2, 2}, which of the following statements is true?
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Which of the following correctly shows the distinction between ∈ and ⊆?
Which of the following correctly shows the distinction between ∈ and ⊆?
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Study Notes
Introduction to Set Theory
- The formal use of "set" in mathematics was established by George Cantor in 1879.
Language and Notation of Sets
- If S is a set, then the notation ( x \in S ) indicates that x is an element of S, while ( x \notin S ) signifies that x is not an element of S.
- Set-roster notation lists all elements within braces, e.g., ( {1, 2, 3} ).
- To describe large or infinite sets:
- ( {1, 2, 3, \ldots, 100} ) symbolizes integers from 1 to 100.
- ( {1, 2, 3, \ldots} ) represents all positive integers.
Examples of Set-Roster Notation
- Sets A, B, and C all contain the same elements: 1, 2, and 3, even if represented differently.
- ( {0} ) is a set containing one element (0), whereas 0 alone is a number, not a set.
- The set ( {1, {1}} ) contains two elements: the number 1 and the set whose only element is 1.
Special Sets of Numbers
- The symbols commonly used in set notation include:
- ( \mathbb{R} ): Set of all real numbers.
- ( \mathbb{Z} ): Set of all integers.
- ( \mathbb{Q} ): Set of all rational numbers (quotients of integers).
Set-Builder Notation
- A set can be defined by a property ( P(x) ) which elements satisfy, noted as ( {x \in S \mid P(x)} ).
Examples of Sets Using Set-Builder Notation
- A set representing all real numbers between -2 and 5 is ( {x \in \mathbb{R} \mid -2 < x < 5} ).
- A set of integers from -1 to 5 is represented as ( {x \in \mathbb{Z} \mid -1 \leq x < 6} ).
Subsets
- A is a subset of B (written ( A \subseteq B )) if every element of A is also in B.
- If at least one element of A is not in B, then A is not a subset.
- A is a proper subset of B if all elements of A are in B, and there exists at least one element in B not in A.
Examples of Subset Relationships
- Assessing sets A, B, and C:
- ( B \subseteq A ): False, since 0 is not a positive integer.
- ( C ) is a proper subset of ( A ): True, as each element of C is in A, but not vice versa.
- Elements common in C and B: True.
Distinction Between Element and Subset Notation
- ( 2 \in {1, 2, 3} ) confirms 2 is an element of the set.
- ( 2 \subseteq {1, 2, 3} ) is incorrect, as 2 is not a set but an element.
- ( {2} \subseteq {1, 2, 3} ) is correct, as {2} is a set containing 2, which is part of the larger set.
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Description
This quiz explores the fundamental concepts of set theory, including notation, elements, and examples of set-roster notation. It delves into special sets of numbers and their representations. Test your understanding of the language and structure used in set theory.
