Chapter 1 Sets
What is an ordered triple?
(x,y,z) (pg. 9)
What is a set?
A collection of things. These things are called the elements of the set and they can be just about anything (pg. 3)
What does A⊈B mean?
A is not a subset of B. It means there is at least one element in A that's not in B. We could also say that it is NOT true that every element of A is an element of B (pg. 11)
Zermelo-Fraenkel axioms
A set of axioms for set theory. Includes the well ordered principle and the axiom of foundation: no non-empty set X is allowed to have the property X∩x≠∅ for all its elements x. (pg. 31)
What is a finite set?
A set that has a finite number of elements (pg. 3)
What is an infinite set?
A set that has an infinite number of elements (pg. 3)
If A∩B=∅, what's A-B and B-A?
A-B=A and B-A=B (pg. 18 excercise 2c and 2e)
Define ordered pair
An ordered pair is a list (x,y) of two things x and y, enclosed in parenthesis and separated by a comma. The order does matter (1,2) ≠ (2,1) (pg. 8)
Why are sets so important?
Because all of mathematics can be described with sets (pg. 3)
What is the cardinality of a power set?
If A is a finite set, the |ℙ(A)|=2^|A|. A power set has 2 to the "cardinality of the original set" elements. This stems from fact 1.3 (fact 1.4 pg. 15)
Define power set
If A is a set, the power set of A is another set denoted as ℙ(A) and is defined to be the set of all subsets of A. In symbols, ℙ(A)={X : X⊆A} ALL ELEMENTS OF A POWER SET NEED TO BE SETS (pg. 14)
When dealing with expressions dealing with unions and intersections, when MUST we use parenthesis?
If the expression uses both unions AND intersections we MUST use parenthesis. If the expression uses only unions or only intersections, parenthesis are optional (pg 23)
What makes two sets equal? Give an example
If they contain the exact same elements. {2,4,6,8} = {8,6,4,2}(pg. 3)
How many subsets does a finite set, B, have?
If |B|=n, then B must have 2^n subsets because every element can be either inserted or not inserted which means there are two options for every element for a total of 2^n options (fact 1.3 pg. 12)
What is a universal set how do we denote it?
It's kind of like the context a set is in. For example the set of prime numbers, P, is in the universe of the set of natural numbers P⊆ℕ. We denote a universal set with the capital letter U (pg. 19)
What is set builder notation?
It's notation used to describe sets that are too big or complex to list between braces. X={expression: rule} where X is all the values of "expression" that are specified by "rule" e.g. E={2n : n∈Z} read, "E is the set of all things of form 2n, such that n is an element of the integers." (see pg. 5 example 1.1)
What is the cardinality of AxB given sets A and B are finite?
It's the cardinality of A times the cardinality of B (Fact 1.1 pg. 9)
What does ℝxℕ look like?
It's the coordinates in ℝ² that have natural numbers as their Y coordinate (figure 1.2b pg. 9)
What does ℕxℕ look like?
It's the coordinates in ℝ² that have natural numbers as their x and y coordinates (figure 1.2c pg. 9)
What is the cardinality of a set?
It's the number of unique elements a finite set has. AKA it's size. We symbolize cardinality with absolute value bars around the capital letter symbolizing the set (pg. 4)
What is an index set?
It's the set for all the subscripts on our sets (pg. 25)
What does ℝxℝ look like?
It's the set of all points on the 2-dimensional Cartesian plane (figure 1.2a pg. 9)
What is the empty set?
It's the set that has no elements. Symbolized by ∅. ∅={} (pg. 4)
What are Cartesian Powers?
Its the Cartesian product of a set with itself n times (see pg. 10)
Define the complement of a set A.
Let A be a set with a universal set U. The complement of A is Everything in U that isn't in A (pg. 19)
Russell's paradox
Let A={X: X is a set and X∉X} Is A∈A? For a set X, the equation says that X∈A means that X∉X. For X=A, this then means that A∈A means the same thing as A∉A. So if A∈A is true, then it is false; if A∉A is false, then it is true (pg. 31)
Is ∅={∅}?
No, think of it like boxes. An empty box is not the same thing as an box with an empty box inside of it. ∅ has a cardinality of zero while {∅} has a cardinality of one (pg. 4)
Can only numbers be the elements of a set?
No, we can put letters, coordinates, sets, matrices and more inside of sets (see pg. 4)
Cartesian product of n sets
See pg. 9
What is the difference of set A with its complement?
Set A. Because the intersection of A and its complement is the empty set. Therefore we remove no elements from A when finding the difference of A and its complement (pg. 20 exercise 2e)
Indexed sets
Sets that are defined by an index, I. e.g. A_1, A_2, A_3,...,A_n is the sets with index set I={1,2,3,...,n} (pg. 24)
Define subset
Suppose A and B are sets. If every element of A is also an element of B, then we say A is a subset of B, denoted A⊆B.
What is the Cartesian product?
The Cartesian product given two sets, A and B, is another set denoted as AxB and is defined as AxB={(a,b) : a∈A and b∈B} notice that the elements of the Cartesian product of two sets are ordered pairs (pg. 8)
What is the complement of the set of prime numbers P?
The composite numbers and 1 (pg. 20)
What is the difference of sets A and B (A-B)
The difference of sets A and B is the set of all things that are in A but not in B. Denoted A-B={x:x∈A and x∉B}. Also denoted A-B or A/B
What is the intersection of a set A with its complement?
The empty set because any set A will share no elements with its complement set (pg. 20 exercise 2c)
Name some important sets
The empty set, the natural numbers, the integers, the rational numbers and the real numbers (pg. 6)
What is the Cartesian product of any set with the empty set ∅?
The empty set. Because |AxB|=|A|*|B| (exercise 1e from 1.2 pg. 10)
What is the difference between ℝx(ℕxℤ) and ℝxℕxℤ?
The first will have elements of the form (a,(b,c)) : a∈ℝ, b∈ℕ, c∈ℤ and the second will be in the form (a,b,c) : a∈ℝ, b∈ℕ, c∈ℤ (pg. 9 exercises 1h,2f,2g,2h pg. 10)
What is the intersection of sets A and B?
The intersection of A and B is the set of all things in BOTH A and B. Denoted A∩B={x:x∈A and x∈B} (pg. 17)
Compare the Venn diagrams of (A∪B)∩C and A∪(B∩C). Are the parenthesis important?
The parenthesis are essential (pg. 22)
Describe ℙ(ℝ²)
The set containing any ordered pair or collection of ordered pairs in the 2D Cartesian plane. "In addition to containing every imaginable function and every imaginable black and white image, ℙ(ℝ²) contains the full text of every book ever written." (pg. 16)
What are elements?
The things that make up the set. They are separated by commas inside of braces (pg. 3)
What is the union of sets A and B?
The union of A and B is the set of all things that are in A OR in B. Denoted A∪B={x:x∈A or x∈B} (pg. 17 )
What is the union of set A with its complement?
The universal set that A and its complement are in (pg. 20 exercise 2d)
How do we symbolically say an element is part of a set?
Using the ∈ symbol. e.g. If S={a,b,c,d,e...} a∈S read "a is an element of S" or, "a is in S" or, "a in S." We could also say a,b,c∈S (pg. 3)
How do we represent sets?
Usually with uppercase letters e.g. A={1,3,5,7,9...}
Is the empty set, ∅, finite?
Yes and it has a cardinality of zero (google)
Can a set be a subset of itself?
Yes, every element in A is in A. e.g. {2,3,7}⊆{2,3,7} all sets are subsets of themselves (example 1.2 #3 pg. 11)
Form intervals given two numbers a,b∈R : a<b
closed interval [a,b], half open interval (a,b] or [a,b), open interval (a,b), infinite interval (a,oo] (pg. 6)
Go over example 1.3 and make sure you understand all of the statements
pg. 13
Be sure to understand the following: X∪Y=Y∪X and X∩Y=Y∩X, but in general X-Y≠Y-X
pg. 17
Compare the Venn diagrams of A∪B∪C and A∩B∩C
pg. 21
Union and intersection of n FINITE sets using notation similar to sigma notation
pg. 24
Union and intersection of an INFINITE number of sets
pg. 25
Do questions 1.8:9,13
pg. 28
ℙ({∅})=?
{∅,{∅}} (pg. 15)
ℙ(∅)=?
{∅} (a set containing all the subsets of the empty set, namely the empty set pg. 15)
Natural numbers
ℕ={1,2,3,4,5...} (positive whole numbers pg. 6) Sometimes zero is included in the natural numbers
Rational numbers
ℚ={x : x=m/n, where m,n∈ℤ and n≠0) The set of all numbers that can be expressed as a fraction of two integers (pg. 6)
Real numbers
ℝ. It's the set of all real numbers on the number line or negative infinity to infinity (pg. 6)
Let A={(x,x^2):x∈ℝ} What does the complement of A look like?
ℝ²-A (pg. 20)
Integers
ℤ={...,-2,-1,0,1,2,...} positive and negative whole numbers (pg. 6)
If A⊆B, what's A-B?
∅ b/c all elements in A are in B so there's nothing in A that's not in B so A-B is the empty set (pg. 18 exercise 2h)
Why is the empty set a subset of every set? That is, ∅⊆B for any set B?
∅⊈B for any set B means there must be at least one element in ∅ that isn't in B. This clearly is not true because ∅ has no elements therefore ∅⊆B for any set B (fact 1.2 pg. 11)