Questions and Answers
Which statement accurately describes what it means for set A to be a subset of set B?
What does the symbol $A subseteq B$ signify?
If set A is not a subset of set B, which of the following conclusions can be drawn?
Which of the following statements correctly represents that set A is a subset of set 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?
Signup and view all the answers
Which of the following statements is true regarding subsets A, B, and C?
Signup and view all the answers
Which condition must be met for C to be classified as a proper subset of A?
Signup and view all the answers
If B = {n ∈ Z | 0 ≤ n ≤ 100}, which statement about the set is true?
Signup and view all the answers
What is the relationship between sets B and C?
Signup and view all the answers
Which of the following represents a true statement about the subsets?
Signup and view all the answers
What does the notation (a, b) represent?
Signup and view all the answers
Under what condition are the ordered pairs (a, b) and (c, d) considered equal?
Signup and view all the answers
Why is the notation for ordered pairs considered simpler than set notation in this context?
Signup and view all the answers
Which of the following statements about ordered pairs is NOT true?
Signup and view all the answers
Which of the following describes what a and b represent in the ordered pair (a, b)?
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}?
Signup and view all the answers
What is the result of the expression {0} = 0?
Signup and view all the answers
For the set defined as $U_n = {n, -n}$, what is $U_0$?
Signup and view all the answers
What does the symbol '...' (ellipsis) indicate when used in set notation?
Signup and view all the answers
Which statement about the axiom of extension is true?
Signup and view all the answers
In the set {1, {1}}, what is true about the element {1}?
Signup and view all the answers
Which of the following notations correctly represents the set of all integers from 1 to 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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.