SETS Flashcards
What is a set
a form of grouping data to form a collection of distinct objects called elements
What is Z(bold) meaning?
Integers
All whole numbers +/- and including 0
What is N(bold) meaning?
Natural Numbers
All whole non-negative numbers, including 0
What is the mathematical enumeration of sets?
A = { Integers X | 1 < X < 6 }
line meaning “such that”
What is the abstraction axiom?
The rule that if property P, there exists a set S of all elements that satisfy property P.
What occurs if nothing meets the property in a set?
An empty set is returned
What are the symbols for an empty set?
Ø or {}
What is the typical notation of a set?
A = { 1, 2, 3, 4, 5, 6 }
What does A ∪ B mean?
Union
Elements belong to set A or Set B. No repetition of elements.
What does A ∩ B mean?
Intersection
Elements that belong to both sets
What does A ⊆ B mean?
Subset
Every element of A is also in B, including when A is equal to B.
What does A ⊂ B mean?
Proper subset
Every element of A is also in B, EXCLUDING when A is equal to B. This means that there exists at least one element that’s in B but not A.
What does A \ B mean?
Set difference
Every element in A but not B
What does a ∈ B mean?
Element
A is an element of B
What does x ∉ A mean?
Not an element
A is not an element of B
What does A ⊆ A ∪ B mean?
A is a subset of A or B
Any element in A is also a part of all elements in A or B
What does A ∩ B ⊆ A
A and B is a subset of A
Any element that is in both A and B (inner part of a venn diagram), will be in A.
What is the cardinality of a set?
{1, 2, 3} = 3
The size of a set (number of elements)
#{} = 0
#{0,1,2,3,4..} = ∞
What is used to denote cardinality of a set?
e.g #{Alice, Bob} = 2
What does P (A) mean?
Powerset
Powerset is a set of ALL subsets of A.
What is an example of a powerset?
if set A = {1,2,3}
P(A) = { {}, {1}. {2}, {3}, {1.2}. {2.3}. {1.3}. {1.2.3}}
what is always included in powerset notation?
Empty set {}
How many elements does P(A) have if A has n elements?
2 ^n
What is #(A ∪ B) equal to?
A + #B - #(A ∩ B)
What does A with a line on top mean?
Complement
Set of all elements ( S ) not in A
= S \ A
What does A = B mean?
Equivalence
A and B have precisely the same elemements
if element in A it must be in B
How to prove A ⊆ B?
Show x ∈ A implies x ∈ B
How to prove A ⊂ B?
Show that x ∈ A and x ∈ B
exists one x where x ∈ B and x ∉ B
How to prove A = B?
Prove A ⊆ B and B ⊆ A
Show x ∈ A if and only if x ∈ B (both directions)
Show x ∈ A implies x ∈ B and x ∈ B implies x ∈ A
How to prove x ∈ A ∪ B ?
Show that x ∈ A or x ∈ B
How to prove x ∈ A\B?
Show that x ∈ A and x ∉ B
How to prove x ∈ A ∩ B ?
Show that x ∈ A and x ∈ B
Definition for if x ∈ A?
Then x ∈ A ∪ B
Definition for if x ∈ A ∩ B?
Then x ∈ A
What is De Morgans law?
A \ (A ∪ B) = (A\B) ∩ (A\C)
A \ (A ∩ B) = (A\B) ∪ (A\C)
What is A ∆ B?
Symmetric difference
Objects that belong to A or B but not their intersection
