Podcast
Questions and Answers
What does the notation A ⊆ B indicate?
What does the notation A ⊆ B indicate?
- No elements of A are in B.
- All elements of A are in B.
- Some elements of A are contained in B.
- Not all elements of A are in B. (correct)
Which of the following examples represents a correct subset relationship?
Which of the following examples represents a correct subset relationship?
- letters that spell 'yes' ⊆ letters that spell 'see'
- negative integers ⊆ positive integers
- vowels ⊆ alphabet (correct)
- odd integers ⊆ even integers
What implication can be drawn from the theorem that states if A ⊆ B and B ⊆ C, then A ⊆ C?
What implication can be drawn from the theorem that states if A ⊆ B and B ⊆ C, then A ⊆ C?
- A is equal to C.
- B is a superset of A.
- C is a subset of A.
- A is a subset of C. (correct)
Which statement is true regarding the empty set (∅)?
Which statement is true regarding the empty set (∅)?
How is cardinality defined for a finite set?
How is cardinality defined for a finite set?
What correctly defines a set?
What correctly defines a set?
What is the power set of a set A?
What is the power set of a set A?
Which of the following sets is correctly defined?
Which of the following sets is correctly defined?
If it is true that A ⊆ B and C ⊆ B, what can be inferred about the relationship between A and C?
If it is true that A ⊆ B and C ⊆ B, what can be inferred about the relationship between A and C?
What does the notation |A| represent?
What does the notation |A| represent?
Which symbol represents an empty set?
Which symbol represents an empty set?
If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}, which statement is true?
If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}, which statement is true?
If C = {x | x is a rational number}, which of the following is true?
If C = {x | x is a rational number}, which of the following is true?
Which description accurately illustrates a subset?
Which description accurately illustrates a subset?
Identify which of the following statements is false regarding sets.
Identify which of the following statements is false regarding sets.
What does the symbol '∈' represent?
What does the symbol '∈' represent?
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Study Notes
Sets and Subsets
- A set is a well-defined collection of objects. Membership can be definitively determined.
- Sets are denoted by uppercase italic letters (e.g., A, B, C).
- Elements of a set are denoted by lowercase letters (e.g., a, b, c).
- The empty set is denoted by ∅ or {}.
- ∈ means "is an element of," ∉ means "is not an element of."
- Sets can be defined by specifying properties their elements must satisfy: A = {a | a is _______} means "the set A is comprised of elements a where a satisfies _______."
Set Notation and Examples
- Sets can be listed using enumeration: {1, 2, 3}.
- Examples of common sets:
- Z⁺: Positive integers
- N: Non-negative integers (positive integers and zero)
- Z: Integers
- Q: Rational numbers (numbers expressible as a/b, where a and b are integers and b ≠ 0)
- R: Real numbers
Subsets
- A ⊆ B means "A is a subset of B." All elements of A are also in B.
- A ⊄ B means A is not a subset of B. This doesn't necessarily mean there are no shared elements.
- Every set is a subset of itself (A ⊆ A).
- The empty set is a subset of every set (∅ ⊆ A).
- If A ⊆ B and B ⊆ A, then A = B.
Venn Diagrams
- Developed by John Venn.
- A visual representation of set relationships; it shows the relationship between sets, not individual elements.
Theorems on Sets
- If A ⊆ B and B ⊆ C, then A ⊆ C (transitive property).
- If A ⊆ B and C ⊆ B, the relationship between A and C is indeterminate (could be A ⊆ C, C ⊆ A, or neither).
Set Cardinality and Power Sets
- A finite set A has n distinct elements; n is the cardinality of A, denoted |A|.
- An infinite set is a set that is not finite.
- The power set P(A) is the set of all subsets of A, including the empty set.
Subset Examples
- Vowels are a subset of the alphabet.
- Letters in "see" are a subset of letters in "yes," which is a subset of letters in "easy."
- Positive integers are a subset of integers, which are a subset of real numbers.
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