Chapter 1: The Real Numbers

What is the contrapositive of a conditional statement p implies q?

Switching the hypothesis and conclusion of a conditional statement and negating both. So the contrapositive of p implies q is not q implies not p.

Redo problem 89. What is the Cantor set? Is it open or closed?

The cantor set is closed. This can be proven using Theorems 3.2.3 (an arbitrary union of open sets is open) and 3.2.13 (a set is open iff its complement is closed) (pg. 94)

Theorem 1.4.2: the Archimedean Property

The first part asserts that the naturals have no upper bound. It's proven using a contradiction and invoking the axiom of completeness. Once part (i) is proven, (ii) part two follows by setting x = 1/y (pg. 21)

Theorem 1.6.1: What is the cardinality of the interval (0,1)?

The interval (0,1) is uncountable. This is proved using Cantor's diagonalization method method (see figure on pg. 33) (pg. 32)

Example 1.3.5: Differentiate between the maximum and supremum of a set

The maximum has to be an element of the set while the suprenum doesn't have to be. Sometimes they are the same, and sometimes they are not. (pg. 16)

Example 1.3.5: Differentiate between the minimum and infimum of a set

The minimum has to be an element of the set while the infinum doesn't have to be. (pg. 16)

powerset

The powerset of a set S is the set of all subsets of S. If the cardinality of S is n then the cardinality of its powerset is 2 to the n.

Give some examples of countable sets

The set of evens, the set of odds, and the set of rationals

Problem 1.4.1(c): Is the set of irrational numbers closed under addition? Under multiplication?

The set of irrationals is not closed under addition nor is it closed under multiplication. √2 and -√2 are both irrational, but their sum √2+(-√2) = 0 is not irrational. Likewise their product √2*(-√2) = -2 is not irrational. (pg. 24)

Real numbers

The set of numbers between negative and positive infinity.

Integers

The set of numbers {..., -2, -1, 0, 1, 2, ...}

Natural numbers

The set of numbers {1, 2, 3, ...}

What is the null set?

The set with no elements (pg. 5)

Theorem 1.5.8: If we have a set of countable sets A₁, A₂, A₃, ... A_m, what can we say about their union? A₁∪A₂∪A₃∪...∪A_m?

The union of any number, even an infinite number, of countable sets is countable (pg. 29)

De Morgan's Laws

These tell us how to find the complement of the intersection of two sets and the complement of the union of two sets.' (pg. 7)

Corollary 1.4.4: the density of the irrationals in the reals

See problem 1.4.5 for proof (pg. 22)

Redo problem 1.2.3

(pg. 11)

Redo problem 1.2.4

(pg. 11)

Define image and preimage

(pg. 12)

Redo problem 1.2.6

(pg. 12)

Definition 1.3.1: bounded above, upper bound, bounded below, lower bound

(pg. 15)

Draw a picture to explain what the supA and infA of a set A are

(pg. 15)

Definition 1.3.4: What is a maximum of a set A? What is a minimum of a set A?

(pg. 16)

Example 1.3.6: Why does the axiom of completeness not apply to the set of rationals?

(pg. 17)

Problem 1.3.1: Prove lemma 1.3.8 but for the greatest lower bound

(pg. 18)

Redo Problem 1.3.10

(pg. 19)

Redo Problem 1.3.6

(pg. 19)

What is the cut property?

(pg. 19)

Redo problem 1.4.2

(pg. 24)

Redo problem 1.4.8

(pg. 24)

Definition 1.5.1: What is a one-to-one function? What is an onto function?

(pg. 25)

Example 1.5.4: Show that (-1, 1)~ℝ

(pg. 26)

Theorem 1.5.6: Show that ℚ is countable while ℝ is uncountable

(pg. 27)

Absolute value function

(pg. 8)

How do we prove that two real numbers are equal? Theorem 1.2.6

(pg. 9)

cardinality of a powerset

2 to the n

What is a field?

A field is any set where addition and multiplication are well-defined operations that are commutative, associative, and obey the familiar distributive property a(b + c) = ab + ac. There must be an additive identity, and every element must have an additive inverse. Finally, there must be a multiplicative identity, and multiplicative inverses must exist for all nonzero elements of the field Neither the integers nor the naturals are fields, however, the set {0, 1, 2, 3, 4} is a field when addition and multiplication are computed modulo 5 (pg. 3)

Define an injective function aka one-to-one

A function f: A→B is injective if for every x,y∈A, x≠y implies f(x)≠f(y). It means the function is one to one. In essence, injective means that unequal elements in A always get sent to unequal elements in B

Define a surjective function aka onto

A function f: A→B is surjective if for every b∈B there is an a∈A with f(a)=b (pg. 201) Surjective means that every element of B has an arrow pointing to it, that is, it equals f (a) for some a in the domain of f Note: a function is surjective if and only if its codomain equals its range

Define a bijective function

A function is bijective if it is both injective and surjective

Definition of greatest lower bound aka infimum

A real number s is a greatest lower bound of a set A if it satisfies two properties: (i) s is a lower bound for set A (meaning s ≤ x for all x∈A) (ii) s is the greatest lower bound (meaning given any lower bound b of A, then s≥b) (pg. 15) the greatest lower bound of a set is UNIQUE (prove it) The greatest lower bound is often denoted infA or GLB(A) (pg. 15)

Binomial theorem

Shows us how to expand any binomial of the form (x+y)^n. The coefficients of each term are nCk were 0≤k≤n

Definition 1.3.2: least upper bound aka supremum

A real number s is a least upper bound of a set A if it satisfies two properties: (i) s is is an upper bound for set A (meaning s ≥ x for all x∈A) (ii) s is the smallest upper bound (meaning given any upper bound b of A then s ≤ b) The least upper bound of a set is UNIQUE (prove it) The least upper bound of a set A is often denoted as supA or LUB(A). (pg. 15)

What is an equivalence relation? Give some examples

A relation R on a set A is an equivalence relation if it is reflexive, symmetric and transitive. The "is equal to", "has the same parity as", "has the same sign as", and "has the same parity and sign as" relations are all equivalence relations

What is a set?

A set is any collection of objects. The objects are referred to as the elements of the set (pg. 5)

What is a subset A of a set B?

A subset A of a set B is a set such that every element in A is also in B (pg. 6)

What is an axiom in mathematics?

An accepted assumption to be used without proof

Definition 1.5.5: What is a countable set? What is an unountable set?

An infinite countable set has the same cardinality as the natural numbers, meaning there exists a bijective function between the two sets (pg. 26)

Theorem 1.6.2: What can we say about the mapping between a set A and it's powerset P(A)?

Any mapping from a set to its powerset is NOT surjective. This is proven by contraction by assuming there exists a surjection f. We then construct the set B = {a∈A | a∉f(a)} (the set of elements in A that are mapped to subsets that do not contain themselves). Since there is a surrjection, there exists an x∈A s.t. f(x) = B. The contradiction arises when we ask whether x∈B or x∉B (pg. 34)

How do we denote that two sets A and B have the same cardinality?

A~B (pg. 26)

State whether the following sets are countable or uncountable (i) The set of natural numbers (ii) The set of even (or odd) numbers (iii) The set of real numbers (iv) The set of rational numbers (v) The integers (vi) The set (-1, 1) (vii) The set of irrational numbers (viii) Any interval (a, b) where a < b

C = countable, U = uncountable (i) C (ii) C (iii) U (iv) C (v) C (vi) U (vii) U (viii) U

Example 1.5.3: show that (i) the set of natural numbers N = {1, 2, 3, ... } and the set E = {2, 4, 6, ...} have the same cardinality: N~E (ii) N and the integers Z = {... -2, -1, 0, 1, 2,...} have the same cardinality: N~Z

Construct the bijective functions in the picture for proof (pg. 26)

Axiom of completeness

Every nonempty set of REAL numbers that is bounded above has a least upper bound (pg. 15)

What makes the set of real numbers different from the set of rationals?

Every nonempty set of real numbers that is bounded above has a least upper bound that is real. It is not the case that every set of rationals that is bounded above has a least upper bound that is also rational.

How to prove biconditional statements p iff q

First prove that if p, then Q. Then prove that if Q, then P

domain, range, codomain, etc.

For a function f:A→B, the set A is called the DOMAIN of f. (Think of domain as the set of possible "input values" for f.) The set B is called the codomain of f. The range of f is the set {f(a):a∈A}

Theorem 1.4.3: the Density of Q in R

For every two real numbers a and b with a < b, there exists a rational number r satisfying a < r < b. This asserts that between every pair of distinct real numbers is a rational number. This theorem is often paraphrased by saying that Q is dense in R. Without working too hard, we can use this result to show that the irrational numbers are dense in R as well (pg. 22)

Definition 1.2.3: What is a function? Domain and range

Given two sets A and B, a function from X to Y is rule or mapping that takes each element x∈X and associates with it a single element of Y. In this case, we write f: X→Y. Given and element x∈X, the expression f(x) is used to represent the element of Y associated with x by f. The set X is called the domain of f. The range is not necessarily equal to Y but refers to the subset of Y given by {y∈Y: y = f(x) for some x∈X}.

symmetric difference of two sets

Given two sets A, B the symmetric difference is the union of the sets A-B and B-A

Theorem 1.4.5: There exists a real number whose square is 2

I don't agree with the proof in the book (or I don't follow it). See this proof https://math.stackexchange.com/questions/1415235/prove-the-existence-of-the-square-root-of-2 (pg. 23)

What does the ∈ symbol mean? When is it used?

If an item a is an element of a set A, then we write a ∈ A (pg. 5)

Lemma 1.3.8: a way to prove that an upper bound of a set A is the suprenum of A.

In words, this is saying that given s is an upper bound, s is the least upper bound if and only if any number smaller than s is not an upper bound (pg. 18)

The two properties that the absolute value function satisfies. Triangle inequality

In words: (i) the absolute value of a product of two real numbers is the product of their absolute values. (ii) the absolute value of a sum of two real numbers is less than or equal to the sum of the their absolute values. The second property is called the triangle inequality (pg. 8)

What are the consequences of the axiom of completeness?

Include the Nested Interval Property, the Archimedean Property, the density of Q and I in R, and the existence of irrational square roots.

What does it mean for two sets to be disjoint?

It means their intersection is the empty set. In other words, they have no common elements (pg. 5)

What is the cardinality of the union of two sets, A and B?

It's the cardinality of A plus the cardinality of B minus the cardinality of their intersection.

Are there more rational or irrational numbers?

More irrational numbers since the set of irrationals is uncountable while the set of rationals is countable

Theorem 1.5.7: If B is countable and A ⊆ B, what can we say about A?

See exercise 1.5.1 for proof (pg. 29)

Theorem 1.4.1 (Nested Interval Property)

This is proved by constructing the set of left-handed endpoints of the intervals and using the axiom of completeness to find its least upper bound. From there, you can show that this least upper bound is in each of the nested intervals and so the intersection is non-empty (pg. 20)

Theorem 1.1.1 There is no rational number whose square is prime

This is proven using a proof by contradiction. We assume that there is a rational number whose square is two and derive a contradiction (pg. 1-2)

Theorem: every nonempty subset of real numbers that is bounded below has a greatest lower bound

This is proven using the axiom of completeness and the two properties of a suprenum/infinum (class)

Problem 1.4.5: The set of irrational numbers is dense in R

This is proven using the fact that the set of rationals is dense in R. Given the real numbers a-√2 and b-√2, there exists a rational number r satisfying a < r + √2 < b. But since a rational number plus an irrational number is irrational, r + √2 is irrational. So given any real numbers a < b, there exists an irrational number t satisfying a < t < b (pg. 24)

Use a truth table to show that a conditional statement and its contrapositive are logically equivalent

This means that to prove p implies q we can prove not q implies not p

nCr

This represents the number of subsets of cardinality r that can be made from a set of cardinality n (review)

Schröder-Bernstein theorem

This tells us that if we can find injections f:A→B and g:B→A then there exists a bijection h:A→B. Thus A~B. Thus to prove two sets A and B have the same cardinality, we need only find an injection from A to B and an injection from B to A (review)(pg. 32)

Outline for proof by contradiction

To prove a statement P is true using proof by contradiction, we assume it's not true and then show that this leads to something absurd (like 1 is even or something pg. 9)

Outline for proof by induction. Proposition: S1,S2,S3,S4,...Sn are true (∀n ∈ N,Sn).

We first prove S1 is true (basis case). Then we prove that given any integer k greater than one, the statement Sk⇒Sk+1 (induction step). It then follows by induction that every Sn is true (pg. 10)

How do we prove that two sets, A and B, are equal?

We show that A is a subset of B and B is a subset of B: given x∈A, show x∈B given x∈B, show x∈A (pg. 6)

Definition 1.5.2: How do we show that two sets have the same cardinality?

We show that there exists a one to one and onto correspondence between the two sets. In other words, construct a bijective function between the two sets (pg. 26)

When is a an item x an element of the union of two sets A and B?

When it's an element of at least one of the sets (pg. 5)

When is an item x an element of the intersection of two sets A and B?

When it's an element of both sets (pg. 5)

What is the complement of a set A?

When we have a subset A of a set R, the complement of A is the set of elements that are in R but not in A (pg. 6)

Problem 1.4.1(a): Are the set of rational numbers closed under addition and multiplication?

Yes, given a,b∈ℚ, a+b∈ℚ and ab∈ℚ. This is not too difficult to prove (pg. 24)

Given a function f and a subset A of its domain, what is the range of f over the set A, f(A)?

f(A) = {f(x): x∈A} (problem 1.2.7)

T/F: zero is in the natural numbers

false

Redo problem 1.2.10

pg. 13

Redo problem 1.2.12

pg. 13

Redo problem 1.5.1

pg. 29

Prove the triangle inequality

problem 1.2.6b

Example 1.3.7. Given a set A and supA, what is the suprenum of the set c + A = {c + a: a ∈A}?

sup(c+A) = c + sup(A) (pg. 17)

rational numbers

the set of numbers that can be written as a ratio of two integers. This includes both positive and negative numbers (pg. 3)