Analysis

Question 1

Abbot Chapter 1.2 The Real Numbers?

What is a field? When can we treat Q as a field?

Answer 1

Q = {all fractions /q where p and q are integers with q != 0}

A field is a set where:

1. Where addition and multiplication are well-defined operations that are commutative (a + b = b + a), associative ( a + (b + c)) = ((a + b) + c) and obey familiar distributive properties (a(b +c)) = (ab + ac)
2. There exists an additive identity, and every element has an additive inverse
3. There exists a multiplicative identity, and every element (except 0) has a multiplicative inverse.

Question 2

Abbot Chapter 1.2 The Real Numbers?

What do we mean when two sets are disjoint?

Answer 2

Question 3

Abbot Chapter 1.2 The Real Numbers?

How do we use the union operator for infinite collections of sets?

Answer 3

Question 4

Abbot Chapter 1.2 The Real Numbers?

Why is the intersection of infinitely many sets the empty set?

Answer 4

• As for any given natural number m (in this case), for m to be an element of each set A_n for all A_n in the sets.

But given that this is infinitely many sets m ∉ A_m+1 given that A_m+1 = {m+1, m+2,m+3…}.

Therefore, no such m exists and the intersection is empty.

Question 5

Abbot Chapter 1.2 The Real Numbers?

What are De Morgan’s laws?

Answer 5

Question 6

Abbot Chapter 1.2 The Real Numbers?

What is the Triangle Inequality?

Answer 6

Question 7

Abbot Chapter 1.2 The Real Numbers?

How do you prove if and only if statements?

Answer 7

Question 8

Abbot 1.3 Axiom of Completeness
What is the Axiom of Completeness?

Answer 8

Every nonempty set of real numbers that is bounded above has a least upper bound

Question 9

Abbot 1.3 Axiom of Completeness
What are the definitions of bounded above, upper bound and least upper bound?

Answer 9

• Part two of the definition of least upper bound is to show that every upper bound heading towards infinity is greater than s, thus s is the start of the upper bounds - thus the least upper bound

Question 10

Abbot 1.3 Axiom of Completeness
What is the definition of the maximum of a set?

Answer 10

Question 11

Abbot 1.3 Axiom of Completeness

Alternative definiton of the Least Upper Bound?

Answer 11

Question 12

Abbot 1.4 Consequences of Completeness

What is Theorem 1.4.1, the nest interval property?

Answer 12

PROOF in short:

To show that the intersection is non-empty
1. Need to consider the lower bounds of a_n.
2. b_n will serve as upper bounds for each n given they are nested. So we can justify setting x = supA (where A is the set of all a_n.
3. As x is an upper bound of a we know that a_n<= x. but we know that each b_n is an upper bound for A. Thus as x is the least upper bound it imples x<= b_n
4. All a_n < x < b_n which means x is in I_n for every choice of n in N. So intersection is non-empty

Question 13

Abbot 1.4 Consequences of Completeness

What is the Archimedean Property of R?

Answer 13

Question 14

Abbot 1.4 Consequences of Completeness

What is the Density of Q in R? What corollary follows from this?

Answer 14

Corollary 1.4.4. Given any two real numbers a<b, there exists an irrational number t satisfying a<t<b.

Proof of the Density property in short:
1. We need to pick some m and n such that a < m/n < b. So we initially pick a value of n such that consecutive increments of 1/n would put us outside the a to b range.
2. Using the Archimedean Property, we can pick n ∈ N large enough so that 1/n < b-a
3. Rearranging the initially inequality, we get an < m < bn. Selecting the smallest value of m ∈ Z, we want to find m-1 < na < m. (with na <m –> a <m/n)
4. m <= na+1 –> m<= n(a+1/n) (which by rearranging Archimedean property and substituting we get m<=n(n-1/n) +1 –> m< bn
5. as m<bn –> m/n < b we have a <m/n < b

Question 15

Abbot 1.4 Consequences of Completeness

What is the Theorem regarding the existence of a root of 2?

Answer 15

Question 16

Abbot 1.5 Cardinality

What is Cardinality in mathematics generally?

Answer 16

The size of a set

Question 17

Abbot 1.5 Cardinality

What is the definiton of a one-to-one function?

Answer 17

Question 18

Abbot 1.5 Cardinality

What is the definition for set A having the same cardinality as set B?

Answer 18

Question 19

Abbot 1.5 Cardinality

What is the definition of a countable set?

Answer 19

Question 20

Abbot 1.5 Cardinality

Is Q and R countable?

Answer 20

Question 21

Abbot 1.5 Cardinality

Theorem around A subset of B if B is countable

Answer 21

Question 22

Abbot 1.5 Cardinality

Theorem around the union of countable sets?

Answer 22

Question 23

Abbot 1.6 Cantor’s Theorem

What do we know about the countability of (0,1)

Answer 23