Introduction to Set Theory

Questions and Answers

What does the notation x ∈ S represent?

x is an element of S (correct)

S is a subset of x

S does not contain x

x is not an element of S

Which of the following represents a set containing only the integer 0?

{0, 1}

{0, 0}

{0} (correct)

How many distinct elements are there in the set {a, {a, b}, {a}}?

2 (correct)

What does the expression Uₙ = n, -n represent for a nonnegative integer n?

A set that includes n and its additive inverse Signup and view all the answers

How can infinite sets be denoted using set-roster notation?

{1, 2, 3, ...} Signup and view all the answers

Which notation correctly represents the set of all integers?

ℤ Signup and view all the answers

In set-builder notation, how is a new set defined?

By specifying a property P that elements of S must satisfy Signup and view all the answers

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}?

They are all equal sets with identical elements Signup and view all the answers

Which of the following represents an open interval of real numbers between -5 and 1?

{𝑥 ∈ 𝑅 | -5 < 𝑥 < 1} Signup and view all the answers

If A = {𝑥 ∈ 𝑍 | 3 ≤ 𝑥 < 8} and B = {𝑥 ∈ 𝑍 | 0 ≤ 𝑥 ≤ 7}, which statement is true?

A is a proper subset of B. Signup and view all the answers

Which of the following correctly describes the set {𝑥 ∈ 𝑍 | -1 ≤ 𝑥 < 6}?

{-1, 0, 1, 2, 3, 4, 5} Signup and view all the answers

What does the notation A ⊆ B imply?

Every element in A is also an element of B. Signup and view all the answers

Which of the following statements is false regarding the set C = {100, 200, 300, 400, 500}?

C contains elements that are not positive integers. Signup and view all the answers

Which set represents integers that are not positive and are greater than or equal to -4 and less than or equal to 0?

{-4, -3, -2, -1, 0} Signup and view all the answers

If A = {2, 2, 2} and B = {2, 2, 2}, which of the following statements is true?

C ⊆ B is true, where C = {2}. Signup and view all the answers

Which of the following correctly shows the distinction between ∈ and ⊆?

2 ∈ {1, 2, 3} and {2} ⊆ {1, 2, 3} Signup and view all the answers

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.