Chapter 2 : Set Theory

Question 1

What is a set?

Answer 1

1. Well-defined collection of distinct objects

• Often have similar properties in a set

Question 2

Set are conventionally denoted with what?

Answer 2

1. Capital Letters ( A , B , C )

A = { Orange, White , Black }
B = { 1 , 2 , 3 , 4 , 5 }
C = { {x,y} , {y,z} } ( Set of Sets )

Question 3

List out ways to describe a set

Answer 3

1. List all the elements ( The set is finite )
2. List a property necessary for its membership ( Large finite set of infinite set )

• W = { 5 , 6 , 7 , 8 , 9 }
  R = { X | X is a positive, even integer } ( 2 , 4 , 6 and so on )

Question 4

What is odd number and even number?

Answer 4

1. Odd Number ( 1 , 3 , 5 , 7 , 11 )
2. Even Number ( 2 , 4 , 6 , 8 , 10 )

Question 5

What does set N has?

Answer 5

1. A set of natural numbers

N = { 0 ,1 , 2 , 3 , 4 , 5 , … }
- No Decimal
- Starts with 0
- Positive Values

Question 6

What does set Z has?

Answer 6

1. Set of itegers

Z = { … , -1 , 0 , 1 , … }
- Positive & Negative
- 0

Question 7

What does set Z⁺ has?

Answer 7

1. Set of integers with positive values only

Z⁺ = { 0 ,1 , … }
- Positive
- 0

Question 8

What does Set Q has?

Answer 8

1. Set of rational numbers

Q = { p / q | p ∈ Z , q ∈ Z , q ≠ 0}
- Fractional Numbers

½ , ⅗ , ⅐ , ⅝

Question 9

What is the other name for rational numbers?

Answer 9

1. Fraction Number

Question 10

What does Set Q ( _ at the top of Q ) has?

Answer 10

1. Irrational Number

• PI , √8

Question 11

Is A equal to B ?
A = { 1 , 2 , 3 }
B = { 3 , 1 , 2 }

Answer 11

1. Yes ( A ∈ B )

• Arrangement doesn’t matter
• This is called set equality

Question 12

When will element X is a subset of Y?

Answer 12

1. If every element of X is an element of Y

• It is like a york when drawn as Venn Diagram
• Subset of a set contains any or all of the elements of the set

A = { 1 , 2 }
B = { 1 , 2 , 3 }
A ⊆ B

A = { 1 , 2 }
B = { 1 , 2 }
A ⊆ B

Question 13

Which answer is correct for
A = { 2 } , B = { 2 , 5 } ?
1. A ⊆ B
2. B ⊆ A
3. A ∈ B

Answer 13

1. True
2. False
3. False

Question 14

Which answer is correct for
A = { 1 , 2 , 3 , 4 , 5 }
B = { X | X < 6 and x is a counting number X ∈ N }
1. A ⊆ B
2. B ⊆ A
3. A ∈ B

Answer 14

1. True
2. True
3. False ( A is not an element of B )

A is a set, but B contains only numbers (1,2,3,4,5).
𝐴 ={1,2,3,4,5}
A={1,2,3,4,5}, but B does not contain the entire set
{1,2,3,4,5} as an element.

Question 15

When is X a proper subset of Y ( X ⊂ Y ) ?

Answer 15

1. If and only If every element in X is also in Y
2. There is at least 1 element in Y not in X

Question 16

Is X proper subset of Y?
X = { a , b ,c }
Y = { a , b , c , e }

Answer 16

1. Yes

Question 17

What is the difference between Subset and Proper Subset?

Answer 17

1. A subset of a set contains any or all of the elements of the set
2. A proper subset contain any of the elements of the set but not all of them ( Minimum 1 isn’t the same )

Question 18

What is an empty set? It was denoted by what symbol?

Answer 18

1. Is the set that has no elements at all
2. It was denoted by ∅

• ∅ = { }
  T = ∅

Question 19

List out prime number from 1 to 100

Answer 19

2 , 3 , 5 , 7 , 11
13 , 17 , 19 , 23 , 29
31 , 37 , 41 , 43 , 47
53 , 59 , 61 , 67 , 71
73 , 79 , 83 , 89 , 97

• Numbers that have only 2 factors: 1 and themselves

Question 20

What is a power set?

Answer 20

1. The set of all subsets

Question 21

What is power set denoted by?

Answer 21

1. P(S)

Question 22

What does this symbol ⊂ mean ?

Answer 22

1. Proper Subset

Question 23

What does this symbol ⊆ mean?

Answer 23

1. Subset

Question 24

What does this symbol ∈ mean ?

Answer 24

1. Set Equality

Question 25

How to calculate elements inside the Power Set?

Answer 25

1. 2^n

n - Elements in the set

x = { a , b } ( 2 Element )
n[P(x)] = 2² = 4
P(X) = { ∅ , {a} , {b} , {a,b} }

Question 26

What is the symbol for Union? What is Union?

Answer 26

1. A ∪ B
2. Set that contains those elements that are either in A or B , or in both

Question 27

What is the symbol for Intersection? What is Intersection?

Answer 27

1. A ⋂ B
2. Set that contains those elements in both A and B

Question 28

When occurs disjoint?

Answer 28

1. 2 sets intersection is an empty set

Question 29

What is the symbol of difference ? What is difference ?

Answer 29

1. A \ B
2. Element in set A that are not in set B

• Also called
  The relative complement of B with respect to A

Question 30

What is the symbol of complement? What is complement?

Answer 30

There is a Universal Set U cover set A

1. _A ( _ is above the A )
2. Means element outside of A

Question 31

What is another way to represent complement?

Answer 31

1. A’

Question 32

What is a Venn Diagram?

Answer 32

1. A pictorial representation of set by sets of points in the plane

Question 33

How to represent Venn Diagram
1. Universal Set
2. Other Sets

Answer 33

1. Interior of Rectangle
2. Disk ( Circle ) lying within the rectangle

Question 34

List out the formula to count the number of elements in an Union ( A∪B )

Answer 34

n(A∪B) = n(A) + n(B) – n(A⋂B)

• Sets must be a finite set to use this formula

Question 35

List out the formula to count the number of elements in an Union ( A∪B∪C )

Answer 35

n(A∪B∪C) = n(A) +n(B) +n(C) –n(A ⋂ B) – n(A ⋂ C) – n(B ⋂ C) + n(A ⋂ B ⋂ C)

• Sets must be a finite set to use this formula