Sets

Question 1

An unordered collection of elements.

Answer 1

Set

Question 2

What are two ways that a set can be represented?

Answer 2

1. Roster or Tabular Form
2. Set Builder Notation

Question 3

define set builder notation?

How does it look syntatically?

Answer 3

Specifying the conditions for elements to be in a set

S = { x | some condition(s) }

Question 4

Notation for element x being a member or non member of set S.

Answer 4

x ∈ S

x ∉ S

Question 5

standard set N?

Answer 5

the set of all natural numbers = {1,2,3,4,…..}

Question 6

standard set Z

Answer 6

the set of all integers = {…..,−3,−2,−1,0,1,2,3,…..}

Question 7

Standard set Z⁺?

Answer 7

The set of all positive integers

Question 8

Standard set Q?

Answer 8

The set of all rational numbers;

rational number: a number that can be expressed as a fraction with a non-zero denominator

Question 9

Standard set R?

Answer 9

all real numbers

Question 10

Standard set W?

Answer 10

All whole numbers

Question 11

Cardinality of a set? denoted?

Answer 11

number of elements of the set

|S|

Question 12

Finite set?

Answer 12

A set which contains a definite number of elements

Question 13

Infinite Set?

Answer 13

A set which contains inifinite numbers or elements

Question 14

Subset? how is it denoted?

Answer 14

A set that lies in another set.

S ⊆ T

“set S is a subset of set T”

Question 15

proper subset? denoted?

Answer 15

subset of but with a smaller cardinality

A ⊂ B

Question 16

Universal set? Denoted?

Answer 16

A collection of all elements in a particular context or application.

big U. “like an umbrella”

Question 17

Empty set or a null set? Denoted?

Answer 17

A set containing no elements?

Ø

Question 18

A singleton set or unit set? Denoted?

Answer 18

A set containing only one element.

{ s }, where s is the single element.

Question 19

Equal sets?

Answer 19

Two sets that contain the same elements.

Question 20

Equivalent sets?

Answer 20

Two sets that have the same cardinality

Question 21

Overlapping sets?

Answer 21

Two sets that have atleast one common element.

Question 22

disjoint set?

Answer 22

Two sets that do not have atleast one element in common.

Question 23

List the types of set operations.

Answer 23

1. Set Union
2. Set Intersection
3. Set Difference
4. Complement of Set
5. Cartesian Product

Question 24

Set Union? Denote? Venn-Diagram?

Answer 24

combining the elements of two sets

A ∪ B = {x | x∈A or x∈B}

Question 25

Set Intersection? Denoted? Venn-Diagram?

Answer 25

The common elements of two sets.

S ∩ T = { x | x∈S and x∈T }

Question 26

Set difference? Denoted? Venn Diagram?

Answer 26

denoted by A - B, contains the elements that are in set A but not in set B

Question 27

Complement of a set? Denoted?

Answer 27

The elements that are not contained in the set.

A’ = ( U - A ), where U is a universal set which contains all objects

Question 28

What is a cartesian Product / Cross Product? give an example.

Answer 28

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)}

Question 29

Power Set?

Answer 29

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}}

Question 30

What is the power set of an empty set?

Answer 30

an empty set

|P({∅})| = 2⁰ = 1

Question 31

Partitioning of a Set?

Answer 31

A collection of disjoint subsets of the original set.

must fulfill 3 conditions:

1. None of the partition sets can be empty sets
2. The union of the partition sets must equal the entire original set
3. The partition sets cannot overlap

Question 32

Bell numbers? denoted?

Answer 32

give the count of the number of ways to partition a set.

B_n, were n is the cardinality of the set.

Example:

S={1,2,3}, S={1,2,3}, n=|S|=3

1. ∅,{1,2,3}
2. {1},{2,3}
3. {1,2},{3}
4. {1,3},{2}
5. {1},{2},{3}

Therefor B₃ = 5

Question 33

{ x | x is an integer, 1 <= x <= 2 }, how does this read?

Answer 33

the set of all x such that x is an integer between 1 and 2 (inclusive)