SETS

Question 1

What is a set

Answer 1

a form of grouping data to form a collection of distinct objects called elements

Question 2

What is Z(bold) meaning?

Answer 2

Integers
All whole numbers +/- and including 0

Question 3

What is N(bold) meaning?

Answer 3

Natural Numbers
All whole non-negative numbers, including 0

Question 4

What is the mathematical enumeration of sets?

Answer 4

A = { Integers X | 1 < X < 6 }

line meaning “such that”

Question 5

What is the abstraction axiom?

Answer 5

The rule that if property P, there exists a set S of all elements that satisfy property P.

Question 6

What occurs if nothing meets the property in a set?

Answer 6

An empty set is returned

Question 7

What are the symbols for an empty set?

Answer 7

Ø or {}

Question 8

What is the typical notation of a set?

Answer 8

A = { 1, 2, 3, 4, 5, 6 }

Question 9

What does A ∪ B mean?

Answer 9

Union

Elements belong to set A or Set B. No repetition of elements.

Question 10

What does A ∩ B mean?

Answer 10

Intersection

Elements that belong to both sets

Question 11

What does A ⊆ B mean?

Answer 11

Subset

Every element of A is also in B, including when A is equal to B.

Question 12

What does A ⊂ B mean?

Answer 12

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.

Question 13

What does A \ B mean?

Answer 13

Set difference

Every element in A but not B

Question 14

What does a ∈ B mean?

Answer 14

Element

A is an element of B

Question 15

What does x ∉ A mean?

Answer 15

Not an element

A is not an element of B

Question 16

What does A ⊆ A ∪ B mean?

Answer 16

A is a subset of A or B

Any element in A is also a part of all elements in A or B

Question 17

What does A ∩ B ⊆ A

Answer 17

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.

Question 18

What is the cardinality of a set?

Answer 18

{1, 2, 3} = 3

The size of a set (number of elements)

#{} = 0
#{0,1,2,3,4..} = ∞

Question 19

What is used to denote cardinality of a set?

Answer 19

e.g #{Alice, Bob} = 2

Question 20

What does P (A) mean?

Answer 20

Powerset

Powerset is a set of ALL subsets of A.

Question 21

What is an example of a powerset?

Answer 21

if set A = {1,2,3}
P(A) = { {}, {1}. {2}, {3}, {1.2}. {2.3}. {1.3}. {1.2.3}}

Question 22

what is always included in powerset notation?

Answer 22

Empty set {}

Question 23

How many elements does P(A) have if A has n elements?

Answer 23

2 ^n

Question 24

What is #(A ∪ B) equal to?

Answer 24

A + #B - #(A ∩ B)

Question 25

What does A with a line on top mean?

Answer 25

Complement

Set of all elements ( S ) not in A
= S \ A

Question 26

What does A = B mean?

Answer 26

Equivalence

A and B have precisely the same elemements
if element in A it must be in B

Question 27

How to prove A ⊆ B?

Answer 27

Show x ∈ A implies x ∈ B

Question 28

How to prove A ⊂ B?

Answer 28

Show that x ∈ A and x ∈ B
exists one x where x ∈ B and x ∉ B

Question 29

How to prove A = B?

Answer 29

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

Question 30

How to prove x ∈ A ∪ B ?

Answer 30

Show that x ∈ A or x ∈ B

Question 31

How to prove x ∈ A\B?

Answer 31

Show that x ∈ A and x ∉ B

Question 31

How to prove x ∈ A ∩ B ?

Answer 31

Show that x ∈ A and x ∈ B

Question 32

Definition for if x ∈ A?

Answer 32

Then x ∈ A ∪ B

Question 33

Definition for if x ∈ A ∩ B?

Answer 33

Then x ∈ A

Question 34

What is De Morgans law?

Answer 34

A \ (A ∪ B) = (A\B) ∩ (A\C)
A \ (A ∩ B) = (A\B) ∪ (A\C)

Question 35

What is A ∆ B?

Answer 35

Symmetric difference

Objects that belong to A or B but not their intersection