Chapter 3: Sets

Question 1

What is the definition of a set and element?

Answer 1

• A set is an unordered collection of distinct objects, which are called elemets
• If x is an elemet of set S we write x ∈ S. This reads “X in S”.

Question 2

What is the set N and Z?

Answer 2

• N = Set of natural numbers
  
  • {1,2,3,….}
• Z = Set of integers {…., -3,-2,-1,0,1,2,3,….}

Question 3

How do you denote a set without any elements?

Answer 3

∅, and a set containing an empty set is {∅}.

Question 4

Set notation and Set-building Notation?

Answer 4

• {elements: conditions used to generate the elements}

OR

• {elements ∈ S: conditions used to generate the elements} –>Where S is some larger set in which the condiitons are restricting

Question 5

What is the set N₀?

Answer 5

the set {0,1,2,3,…}
also denoted as {|n|:n ∈ X}

Question 6

What is the special set Q?

Answer 6

The Set Q is the set of rationale numbers

Q = { (a/b) : a,b ∈ Z, b≠0}

Question 7

What is the set R?

Answer 7

• Set of Real Numbers
• Hard to define but it is basically any numbers that you can write down with or without decimal points.

Question 8

What is the set R^2?

Answer 8

Can be used to represent the x,y plane, which is the set of ordered pairs of real numbers

R² = {(x,y}: x ∈ R and y ∈ R}

Question 9

What is the set for the unit circle?

Answer 9

for a circle of radius 1 centered at the origin it is contained within the set of R^{2 and can be definied as the following:}

S¹= {x,y ∈ R²: x²+ y² = 1}

Question 10

How do you write a close interval as a set?

Answer 10

[a,b] = {x ∈ R: a ≤ x ≤ b}

Question 11

How do you denote that A is a subset of B?

Answer 11

Suppose A and B are a sets. If every elemet in A is also an element of B, then A is a subset of B, which is denoted as A⊆B

Question 12

What sets are Natural numbers a subset of?

Answer 12

N⊆Z⊆Q⊆R⊆C

where the set C is complex numbers

Question 13

How for every set B, is ∅ always a subset of it?

Answer 13

• As it meets the definition of a subset
• As there are no elements in the set ∅, it would be true to say “for and x∈ ∅, x is a purple elephant that speaks German”
• It’s vacuously true as you certainly cant disprove it right. as you cant present any element in the set ∅ that is not a purple elephant that speaks German
• If I can swap in any statement here to make it true then also the statement X ∈ ∅ then x∈ B is also true and by definition ∅⊆B.

Question 14

What is a proper subset?

Answer 14

• IF A = B, then A ⊆ B
• In the case that A ⊆ B and A ≠B, we can say that A is a proper subset of B
• This is denoted as A⊂ B

Question 15

What is the standard template of a “subset” direct proof?

Answer 15

• Proposition A⊆B
• Proof Assume x ∈ A
  * «An explanation of what x ∈ A means»
  * apply algebra, logic, techniques
  * Oh hey, that what x ∈ B means>
• Therefore x ∈ B

Since x ∈ A implies that x ∈ B, it follows that A⊆B

Question 16

Proof:
It is the case that

{ n ∈ Z:12|n} ⊆ {n ∈ Z :3|n}

Question 17

Proof:
Let A ={1,-3} and B {x ∈ R, x^{3 - 3x2 - x +3 =0}}

Question 18

How do you prove that A = B and what is the standard form of the proof?

Answer 18

• To say A= B both must share the exact same elements
• There for you must prove that A⊆B and also B ⊆A

Question 19

What are the union and intersection of sets?

Answer 19

• The union of sets A and B is the set A ∪ B = {x : x ∈ A or x ∈ B}
• The intersection of sets A and B is the set A ∩ B = {x : x ∈ A and x ∈ B}
• Likewise, if A₁, A₂, A₃, . . . , An are all sets, then the union of all of them
  is the set A₁ ∪ A₂ ∪ · · · ∪ A_{n = {x : x ∈ Ai for some i}. This set is also denoted as (big U)i=1ⁿ Ai}
• • Likewise, if A₁, A₂, A₃, . . . , An are all sets, then the intersection of all of
    them is the set A₁ ∩ A₂ ∩ · · · ∩ A_n = {x : x ∈ A_i for all i}. This set is also
    denoteds (big N)_i=1ⁿ A_i

Question 20

What is the Subtration and Complements of a set?

Answer 20

Assume A and B are sets and “x∉B” means that x is not an element of B.
* The subtraction of B from A is A \ B = {x : x ∈ A and x∉B}
* * If A ⊆ U, then U is called a universal set of A. The complement of A in U is A^c = U \ A.

Question 21

What is important to note about subtrations of sets versus the intersection/union of sets?

Answer 21

• Note that A ∪ B = B ∪ A and A ∩ B = B ∩ A are always true, but A \ B = B \ A is rarely true.

Here are some examples:

1. Let A be the set of odd integers and B be the set of even integers
   *
   2.IF Z is the universal set

Question 22

What are Power Sets and Cardinality?

Answer 22

• The Power set of A is P(A) = {X: X⊆A}
  
  • The elements of P(A) are all sets X that satisfy the conditions of being a subset of A
  • So this would be all permutation and variations of that set including the empty set ∅
• The Cardinality of A is the number of elemets in A and is noted |A|
  
  • This only could the elements of that set, therefore if you had a set of three separate sets (containing 5 elements) it would only have a cardinality of 3.

Question 23

What is the Cartesian product of a set?

Answer 23

Assume that A and B are sets

The Cartesian product of A and B is A x B = {(a,b} = a ∈ A and b ∈ B}

• It is like a way to “multiply” sets which creates an order pair of all elements where elements of A would be on the x axis and elements of B would be on the y axis.

Question 24

What are the set Operators?

Answer 24

• Union and Intersection
• Subtration and Complements
• Power Sets and Cardinality
• Cartesian Products

Question 25

Proof: Suppose A and B are sets. If P(A) ⊆ P(B), then A ⊆ B?

Question 26

What is De Morgan’s Laws?

Answer 26

Suppose A and B are subsets of a universal set U Then.

• (AuB)^c= A^c n B^c
• (AmB)^c= A^c u B^c

Question 27

What is the proof of De Morgan’s laws?

Answer 27

Question 28

How do you prove a ∈ A under the following generic set notation?

A = {x ∈ S:P(x)}

Answer 28

Considering the set A = {x ∈ S:P(x)}

where P(x) is some condition n x (say A = {x ∈ Z : 6 | x}, then P(x) is the ocnditions “6|x”.

Given a set of this form, if you are presented with a specific a and you wish to prove that a ∈ A then you must show that

i) a ∈ S and ii) P(a) is true

Question 29

Proof: Let A = {(x,y) ∈ Z x N: x ≡ y(mod5)}, Then (17,2 ) ∈ A.

Question 30

What is a family of sets and how do you denote the union and intersection of these sets?

Answer 30

denoted as F, a set is referred to as a family of sets if all elements of the set are sets themselves

Question 31

Proof: In the case that

{n ∈ Z: 12|n} = {n ∈ Z: 3|n} ∩ {n ∈ Z: 4|n}

And what interesting Proposition does this give rise too? –> might need separate bullet points

Question 32

How can π/4 be defined as an equation?

Answer 32

1 -(1/3)+(1/5)-(1/7)+(1/9)-……

Question 33

Given that 3²=9 are two mathematical objects that are equal (and in term of sets equal)

What is the trignometric set that the following cartesian set equal to:

{(x,y}: x² + y² = 1

Answer 33

{(cos(t),sin(t)) : 0 <= t <= 2π}

Question 34

What is the Cardinality of the Power Set?

Answer 34

If |A|=n, then |P(A)|=2<sup.n</sup>.

Question 35

Proposition: Given any A ⊆ {1,2,3,…100} for which |A|=10, there exist two different subsets X⊆ A and Y⊆A for which the sum of the elements in X is equal to the sum of the components in Y

Answer 35

• For example 10 random numbers from 1-100 could be
  
  • {6,23,30,39,44,46,62,73,90,92}
• And sure enough there are two subsets X and Y which work If we let X = {6,23,46,73,90} and Y = {30,44,73,91}
• Here elements in both sets sum to 238.
• This would still apply if we added/subtracted any other number from the randomly selected to set to the set of X and Y
• To check if this proposition is “guaranteed” we need to apply the Pigeonhole Principle: