Mathematics Section 2.2: Language of Sets

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Questions and Answers

Which statement accurately describes what it means for set A to be a subset of set B?

All elements of B are in A.

A and B must contain the same elements.

Every element of A is also an element of B. (correct)

There is at least one element in A that is not in B.

What does the symbol $A subseteq B$ signify?

A contains all elements of B.

At least one element of A is not in B. (correct)

All elements of B are in A.

A is equal to B.

If set A is not a subset of set B, which of the following conclusions can be drawn?

There is at least one element in A that is not part of B. (correct)

A is the same as B.

B contains all elements of A.

Every element of B is in A.

Which of the following statements correctly represents that set A is a subset of set B?

If $x otin A$, then $x otin B$. Signup and view all the answers

Which of the following phrases is an alternative way to express that A is a subset of B?

B contains A. Signup and view all the answers

Which of the following statements is true regarding subsets A, B, and C?

C is a subset of B Signup and view all the answers

Which condition must be met for C to be classified as a proper subset of A?

There must be at least one element in A not present in C Signup and view all the answers

If B = {n ∈ Z | 0 ≤ n ≤ 100}, which statement about the set is true?

Zero is included in B Signup and view all the answers

What is the relationship between sets B and C?

C and B have at least one element in common Signup and view all the answers

Which of the following represents a true statement about the subsets?

C is equal to itself Signup and view all the answers

What does the notation (a, b) represent?

An ordered pair indicating the position of a and b. Signup and view all the answers

Under what condition are the ordered pairs (a, b) and (c, d) considered equal?

a = c and b = d must both hold true. Signup and view all the answers

Why is the notation for ordered pairs considered simpler than set notation in this context?

It reduces the complexity by requiring fewer symbols. Signup and view all the answers

Which of the following statements about ordered pairs is NOT true?

The notation (a, b) describes an equivalent set representation. Signup and view all the answers

Which of the following describes what a and b represent in the ordered pair (a, b)?

a is the first element and b is the second element. Signup and view all the answers

Which statement accurately describes the relationship among the sets A = {1, 2, 3}, B = {3, 1, 2}, and C = {1, 1, 2, 3, 3, 3}?

A, B, and C have the same elements and represent the same set. Signup and view all the answers

What is the result of the expression {0} = 0?

False, they are not equivalent. Signup and view all the answers

For the set defined as $U_n = {n, -n}$, what is $U_0$?

{0, 0} Signup and view all the answers

What does the symbol '...' (ellipsis) indicate when used in set notation?

It represents a continuation of elements. Signup and view all the answers

Which statement about the axiom of extension is true?

It affirms that a set is defined solely by its unique elements. Signup and view all the answers

In the set {1, {1}}, what is true about the element {1}?

It is a set containing one element. Signup and view all the answers

Which of the following notations correctly represents the set of all integers from 1 to 100?

{1, 2, 3, ..., 100} Signup and view all the answers

Study Notes

Introduction to Sets

The term set was formalized by Georg Cantor in 1879.
A set is intuitively understood as a collection of distinct elements.

Set Notation

If S is a set, then $x ∈ S$ indicates that x is an element of S, while $x ∉ S$ shows x is not in S.
Set-roster notation represents a set by listing its elements within braces, e.g., {1, 2, 3}.
For large sets, notation like {1, 2, 3,..., 100} indicates elements from 1 to 100.
An infinite set can be represented as {1, 2, 3,...} for all positive integers.
The ellipsis (...) signifies continuation in such notations.

Axiom of Extension

A set is defined solely by its elements, disregarding order or duplicates.

Examples of Sets and Elements

Sets A, B, and C containing elements {1, 2, 3} are equivalent despite different listings.
{0} is not equal to 0; {0} is a set containing one element (0), whereas 0 is just the numeral.
The set {1, {1}} consists of two elements: 1 and a set containing 1.
For each nonnegative integer n, the set $U_n = {n, -n}$ yields: $U_1 = {1, -1}$, $U_2 = {2, -2}$, and $U_0 = {0}$.

Subsets

A set A is a subset of set B, written $A \subseteq B$, if all elements of A are also in B.
The statement $A \nsubseteq B$ occurs if at least one element of A isn’t found in B.

Proper Subset

A set A is a proper subset of B if every element of A is in B and B contains an element not in A.

Evaluating Subset Relationships

Given A = Z+, B = {n ∈ Z | 0 ≤ n ≤ 100}, and C = {100, 200, 300, 400, 500}:
- B ⊆ A is false; zero is in B but not in A.
- C is a proper subset of A, B, and C have common elements.
- C ⊆ B is false, as all elements of C exceed 100.
- C ⊆ C is always true since every set contains itself.

Ordered Pairs

Ordered pairs are denoted as (a, b), specifying the sequence of elements.
Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.

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Description

Explore the foundational concepts of sets in mathematics as introduced by Georg Cantor. This quiz covers the notation and ideas surrounding sets, including examples of elements and collections. Test your understanding of set theory and its applications in mathematics.