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Questions and Answers
Which of the following represents the set of all positive integers?
What is the correct set-builder notation for all odd integers less than 0?
Which of the following is indicated by the notation {y | y ≠ 15}?
What does the Cartesian product A X B produce?
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Which of the following correctly defines the set A for all integers greater than 4?
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Which property demonstrates that the intersection of two sets A and B is also a subset of set A?
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What is the expression for the set difference between set A and set B?
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Which of the following identities is an example of De Morgan's Law?
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What is the complementary identity of the universal set U?
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Which law specifies that both A∩A=A and A∪A=A?
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What method is employed to establish the equality of two sets by proving both A⊆B and B⊆A?
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Which identity states that the union of set A and the intersection of A and B equals set A?
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In a Venn diagram proof, what is a notable drawback when considering a large number of sets?
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What is the process used to prove set relations by showing one set is a subset of another called?
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If A is a subset of B and B is a subset of C, what can be concluded about the relationship between A and C?
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Which set identity is represented by the expression (A∩B) - (B-C)?
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When simplifying sets using the Law of Sets, which type of proof is typically employed?
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In the proof for the identity (AUB') = Α' ∩ B', what does the expression indicate about the element x?
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Which law allows the transformation of (A∩B) ∪ (A∩C) into A∩(B∪C)?
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What is the result of applying the identity A ∩ B' when expressing A - B?
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In the element method for proving set relations, what must be shown for a set to be established as a subset?
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Which of the following defines a singleton set?
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What characterizes the cardinality of a set?
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Which of the following sets is an example of an infinite set?
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Which statement is true about disjoint sets?
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What distinguishes a proper subset from a regular subset?
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The notation Ø represents which type of set?
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Which method represents a set by listing its elements within braces?
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How are equivalent sets defined?
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Study Notes
Set
- An unordered collection of objects
- Each object is called an element or member
- Represented by capital letters (A, B, C)
- Examples:
- A = Set of all the planets in the solar system
- L = Set of all lowercase letters of the alphabet
- D = {1, 2, 3, 4, 5}
- J = 0
- F = {F,A,N,N,Y}
- J= {}
- J = {0}
Sets do not have duplicate elements
- Order does not matter
Cardinality of Set
- The number of elements in the set
- Example: Let C = {yellow, blue, red}. The cardinality of C is 3, |C| = 3
Types of Sets
- Finite set: Contains a definite number of elements. Example: A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
- Infinite set: Contains an infinite number of elements. Example: Z*= {1, 2, 3, 4,...}
- Empty set or null set: An empty set contains no elements. Ø = {}
- Universal set: A set encompassing all elements within a specific context. Example: U set of all animals on earth
- Singleton or unit set: A set with only one element
- Equal set: Two sets containing the same elements
- Equivalent set: Two sets with the same cardinality
- Subset: A set X is a subset of set Y if every element of X is an element of set Y, denoted by "C". Example: A ⊆ B
- Proper subset: A subset where every element of X is an element of Y but X is not equal to Y, denoted by "⊆C"
- Overlapping set: Two sets sharing at least one common element
- Disjoint set: Two sets without any common elements
Two methods of Set representation
-
Roster / Tabular form:
- Elements listed separated by commas and enclosed in braces { }.
- Example:
- F= {Abe, LotLot, Dolor, Anne, Ethel}
- Z= {...,-3,-2,-1,0,1,2,3,...}
- N = {1,2,3,4,5}
-
Set Builder Notation:
- Defines a set by specifying a property shared by its elements.
- General form: {x|P(x)}
- | - read as "such that",
- P(x) - properties of x
- Example: {x | x > 0} - the set of all x's, such that x is greater than 0.
Set Builder Notation Symbols
- E: is an element of
- -: is not an element of
- N: set of all natural numbers
- Z: set of all integers
- Z*: set of all positive integers
- Q: set of all rational numbers, {p/q | p ∈ ℤ, q ∈ ℤ, and q ≠ 0}
- R: set of all real numbers, including both rational and irrational numbers
- R*: set of all positive real numbers
- W: the set of all whole numbers
Examples of Set Builder Notation:
- a. Even integers: {x|x=twice the integer}
- b. Numbers whose square root is an integer: {x| √x ∈ ℤ}
- c. Set A is the set of all 'x' such that 'x' is a natural number between 5 and 10. A = {x | x ∈ ℕ, 5 < x < 10}
- d. Set A is a set of odd positive integers below 11. A = {x ∈ ℕ | x < 11, x is odd}
- e. Set of letters in the word "California". A = {x | x is a letter of the word "California"}
- f. Any value greater than 0. A = {y|y>0}
- g. Any value except 15. A = {y|y ≠ 15}
- h. Any value less than 7. A = {y|y<7}
- i. All integers greater than 4. A = {y ∈ ℤ | y>4}
Ordered Pairs
- Pairs of elements in a specific order enclosed in brackets
- Different orders create different pairs: (a, b) ≠ (b, a)
Cartesian Product
- The set of all ordered pairs formed by elements from two sets A and B, denoted by A X B
- Example: A = {1, 2} and B = {3, 4, 5}, A X B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
SET Identities
- Key relations and properties between sets
Properties for Subset Relations
- (A∩B)⊆A: Inclusion for Intersection
- (A∩B)⊆B:
- A⊆(AUB): Inclusion for Union
- B⊆(AUB):
- (A⊆B)(B⊆C)(A⊆C): Transitive Property of Subsets
Properties of Universal Set and Empty Set
- AUØ=A: Domination Laws
- A∩Ø=Ø:
- A∩A'=Ø: Complementation Laws
- AUA'=U:
- U'=Ø:
- Ø' = U:
Set Identities
- A∩B=B∩A: Commutative Laws
- A∪B=B∪A:
- (A∩B)∩C=A∩(B∩C): Associative Laws
- (A∪B)∪C=A∪(B∪C):
- A∪(B∩C)=(A∪B)∩(A∪C): Distributive Laws
- A∩(B∪C)=(A∩B)∪(A∩C):
- A∩U=A: Identity Law
- A∪U=U: (Intersection and Unition with Universal Set)
- (A')'= A: Double Complement Laws
- A∩A=A: Idempotent Laws
- A∪A=A:
- (A∩B)'=A'∪B': De Morgan's Laws
- (A∪B)'=A'∩B':
- A∪(A∩B)= A: Absorption Laws
- A∩(A∪B) = A:
- A-B=A∩B': Set Difference Law
Methods of Proving Set Identities
- Element method or double inclusion method: Proving A⊆B and B⊆A to demonstrate equality
- Venn Diagram: Visual representation of sets. Useful for a small number of sets, but may lack rigor.
- Membership Table: A truth table-like representation of set membership. Simple but becomes less effective with more sets.
- Proofs For Set Relations: Demonstrating subset relationships using the element method and direct proof style.
- Proofs For Set Identities: Using definitions, proof strategies, and set identities to prove equivalence.
Examples:
- Prove (AUB)-B = A-B using Venn diagrams and membership tables
- Proof of transitive property of subsets: If A⊆B and B⊆C, then A⊆C
- Show that (A UB') = Α' ∩ B' using element method
- Prove (A∩B) - (B-C) = A-B using Laws of Set
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