Sets Flashcards
An unordered collection of elements.
Set
What are two ways that a set can be represented?
- Roster or Tabular Form
- Set Builder Notation
define set builder notation?
How does it look syntatically?
Specifying the conditions for elements to be in a set
S = { x | some condition(s) }
Notation for element x being a member or non member of set S.
x ∈ S
x ∉ S
standard set N?
the set of all natural numbers = {1,2,3,4,…..}
standard set Z
the set of all integers = {…..,−3,−2,−1,0,1,2,3,…..}
Standard set Z+?
The set of all positive integers
Standard set Q?
The set of all rational numbers;
rational number: a number that can be expressed as a fraction with a non-zero denominator
Standard set R?
all real numbers
Standard set W?
All whole numbers
Cardinality of a set? denoted?
number of elements of the set
|S|
Finite set?
A set which contains a definite number of elements
Infinite Set?
A set which contains inifinite numbers or elements
Subset? how is it denoted?
A set that lies in another set.
S ⊆ T
“set S is a subset of set T”
proper subset? denoted?
subset of but with a smaller cardinality
A ⊂ B
Universal set? Denoted?
A collection of all elements in a particular context or application.
big U. “like an umbrella”
Empty set or a null set? Denoted?
A set containing no elements?
Ø
A singleton set or unit set? Denoted?
A set containing only one element.
{ s }, where s is the single element.
Equal sets?
Two sets that contain the same elements.
Equivalent sets?
Two sets that have the same cardinality
Overlapping sets?
Two sets that have atleast one common element.
disjoint set?
Two sets that do not have atleast one element in common.
List the types of set operations.
- Set Union
- Set Intersection
- Set Difference
- Complement of Set
- Cartesian Product
Set Union? Denote? Venn-Diagram?
combining the elements of two sets
A ∪ B = {x | x∈A or x∈B}
Set Intersection? Denoted? Venn-Diagram?
The common elements of two sets.
S ∩ T = { x | x∈S and x∈T }
Set difference? Denoted? Venn Diagram?
denoted by A - B, contains the elements that are in set A but not in set B
Complement of a set? Denoted?
The elements that are not contained in the set.
A’ = ( U - A ), where U is a universal set which contains all objects
What is a cartesian Product / Cross Product? give an example.
Product of n number of sets producing all possible ordered pairs
A={a,b} and B={1,2}
A×B = {(a,1),(a,2),(b,1),(b,2)}
B×A = {(1,a),(1,b),(2,a),(2,b)}
Power Set?
if you have a set S it would be the all the possible subsets of S. so the cardinality of a power set of S is equal to 2 raised to the cardinality S
S = {a,b,c,d}
P(S) =
{{∅},{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}}
What is the power set of an empty set?
an empty set
|P({∅})| = 20 = 1
Partitioning of a Set?
A collection of disjoint subsets of the original set.
must fulfill 3 conditions:
- None of the partition sets can be empty sets
- The union of the partition sets must equal the entire original set
- The partition sets cannot overlap
Bell numbers? denoted?
give the count of the number of ways to partition a set.
Bn, were n is the cardinality of the set.
Example:
S={1,2,3}, S={1,2,3}, n=|S|=3
- ∅,{1,2,3}
- {1},{2,3}
- {1,2},{3}
- {1,3},{2}
- {1},{2},{3}
Therefor B3 = 5
{ x | x is an integer, 1 <= x <= 2 }, how does this read?
the set of all x such that x is an integer between 1 and 2 (inclusive)
