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Questions and Answers
Which statement accurately describes what it means for set A to be a subset of set B?
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?
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?
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?
Which of the following statements correctly represents that set A is a subset of set B?
Which of the following phrases is an alternative way to express that A is a subset of B?
Which of the following phrases is an alternative way to express that A is a subset of B?
Which of the following statements is true regarding subsets A, B, and C?
Which of the following statements is true regarding subsets A, B, and C?
Which condition must be met for C to be classified as a proper subset of A?
Which condition must be met for C to be classified as a proper subset of A?
If B = {n ∈ Z | 0 ≤ n ≤ 100}, which statement about the set is true?
If B = {n ∈ Z | 0 ≤ n ≤ 100}, which statement about the set is true?
What is the relationship between sets B and C?
What is the relationship between sets B and C?
Which of the following represents a true statement about the subsets?
Which of the following represents a true statement about the subsets?
What does the notation (a, b) represent?
What does the notation (a, b) represent?
Under what condition are the ordered pairs (a, b) and (c, d) considered equal?
Under what condition are the ordered pairs (a, b) and (c, d) considered equal?
Why is the notation for ordered pairs considered simpler than set notation in this context?
Why is the notation for ordered pairs considered simpler than set notation in this context?
Which of the following statements about ordered pairs is NOT true?
Which of the following statements about ordered pairs is NOT true?
Which of the following describes what a and b represent in the ordered pair (a, b)?
Which of the following describes what a and b represent in the ordered pair (a, b)?
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}?
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}?
What is the result of the expression {0} = 0?
What is the result of the expression {0} = 0?
For the set defined as $U_n = {n, -n}$, what is $U_0$?
For the set defined as $U_n = {n, -n}$, what is $U_0$?
What does the symbol '...' (ellipsis) indicate when used in set notation?
What does the symbol '...' (ellipsis) indicate when used in set notation?
Which statement about the axiom of extension is true?
Which statement about the axiom of extension is true?
In the set {1, {1}}, what is true about the element {1}?
In the set {1, {1}}, what is true about the element {1}?
Which of the following notations correctly represents the set of all integers from 1 to 100?
Which of the following notations correctly represents the set of all integers from 1 to 100?
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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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