Real Analysis

Question 1

x -> y

Answer 1

if x, then y

Question 2

y -> x

Answer 2

x if y

Question 3

rational number

Answer 3

any number that can be expressed in the form p/q where p and q are integers

Question 4

proof by contradiction

Answer 4

assume there is the OPPOSITE of what you’re claiming and proceed until we reach an unacceptable conclusion, thus, you must conclude the opposite

Question 5

if some number is divisible by 2

Answer 5

some number must be a even number

Question 6

square of an odd number

Answer 6

is an odd number

Question 7

natural numbers (N)

Answer 7

non-negative, non-zero, integers (ex. 1, 2, 3, 4, 5…)

Question 8

Z

Answer 8

integers (ex. -3, -2, -2, 0, 1, 2, 3)

Question 9

Q (rational numbers)

Answer 9

all fractions (p/q) are integers where q \neq 0

Question 10

Q property

Answer 10

has a natural order: given any two rational #’s r and s: r < s, r = s or r > s

Question 11

set

Answer 11

any collection of objects

Question 12

element

Answer 12

objects in a set

Question 13

x ∈ A

Answer 13

x in an element of the set A

Question 14

x ∉ A

Answer 14

x is not an element of the set A

Question 15

A U B

Answer 15

union of A and B

Question 16

x ∈ A U B

Answer 16

x ∈ A or x ∈ B (or both)

Question 17

A ∩ B

Answer 17

intersection of A and B

Question 18

x ∈ A ∩ B

Answer 18

x is an element of A and x is an element of B

Question 19

∅

Answer 19

set with no elements (empty set)

Question 20

A is a subset of B

Answer 20

every element in A is also an element of B (A ⊆ B)

Question 21

A = B

Answer 21

A ⊆ B AND B ⊆ A ( A contains B, B contains A meaning they have exactly the same element)

Question 22

function from A to B

Answer 22

a rule or mapping that takes each element x ∈ A and associates is with a single element in B (f : A -> B)

Question 23

closed set

Answer 23

a set A ⊆ R is closed if the set A contains all of the limit points

Question 24

the well ordering principle

Answer 24

every non empty set of N has a smallest element

Question 25

De morgans law

Answer 25

The complement of the union (U) of two sets is equal to the intersection (∩) of their individual compliments

Question 26

set closure

Answer 26

A U A' (the union of A and it's compliment)

Question 27

compliment of a set (A')

Answer 27

the set of elements not in A

Question 28

supremum

Answer 28

least upper bound

Question 29

infinimum

Answer 29

greatest lower bound

Question 30

a set is **bounded above** if

Answer 30

there exists a real number, m, such that for every element `(x)`in the set, `(x \le m)`

Question 31

the set **A** is **bounded below** if

Answer 31

there exist a *real number*, **n** satisfying n* less than or equal to* a for every a that is an element of **A**

Question 32

the well ordering principle

Answer 32

every non-empty subset of N has a smallest element

Question 33

axiom of completeness

Answer 33

every non-empty set of real numbers that is bounded above has a least upper bound

Question 34

axiom

Answer 34

property of the natural numbers

Question 35

principal of mathematical induction

Answer 35

for each n ∈ N let p(n) be a logical proposition depending on n. If p(1) is true for all k ∈ N we have p(k) -> p(k+1) then p(n) is true for all n ∈ N.

Question 36

Steps for induction

Answer 36

1. Establish p(n) 2. Show that p(1) is true 3. Let k ∈ N show that if p(k) is true then p(k+1) is true. 4. Conclude that by the principal of mathematical induction, p(n) is true for all n ∈ N.

Question 37

bounded above

Answer 37

there exists some M ∈ R such that a ≤ M for all a ∈ A

Question 38

bounded below

Answer 38

there exists some m ∈ R such that m ≤ a for all a ∈ A

Question 39

bounded

Answer 39

there exist m, M ∈ R such that m ≤ a ≤ M for all a ∈ A

Question 40

unbounded

Answer 40

it is either not bounded below or not bounded above (For example, (0, ∞) is bounded below but it is unbounded because it is not bounded above.)

Question 41

How to prove M is the supremum of A

Answer 41

1. Show that M is an upper bound of A. 2. Let a ∈ A. Show that a ≤ M 3. Let N be an upper bound of A. Show that M ≤ N *For this proof to work, it is necessary that A ⊆ R is nonempty.

Question 42

How to prove that m is the infinimum of A

Answer 42

1. Show that m is a lower bound of A. 2. Let a ∈ A. Show that m ≤ a. 3. Let n be a lower bound of A. Show that n ≤ m. *For this proof to work, it is necessary that A ⊆ R is nonempty.

Question 43

M ∈ R is the maximum of the non-empty set A

Answer 43

if M ∈ A and a ≤ M for all a ∈ A

Question 44

m ∈ R is a minimum of the non-empty set A

Answer 44

if m ∈ A and m ≤ a for all a ∈ A

Question 45

If max A exists, what is true about the supremum

Answer 45

sup A = max A.

Question 46

if min A exists what is true about the infinimum (inf)

Answer 46

inf A = min A

Question 47

Archimedean Property

Answer 47

the natural numbers are unbounded

Question 48

If [a1, b1] ⊇ [a2, b2],

Answer 48

then a1 ≤ a2 ≤ b2 ≤ b1.

Question 49

Proving ⋂∞ n=1[an, bn] = {c}.

Answer 49

1. Determine c. 2. Prove that c ∈ ⋂∞ n=1[an, bn]. 3. Prove that if x ∈ ⋂∞ n=1[an, bn], then x = c. 4. Conclude that ⋂∞ n=1[an, bn] = {c}.

Question 50

function

Answer 50

a rule that assigns to every element x in a set X a unique element y in a set Y . We write f : X → Y .

Question 51

Let f : X → Y be a function. We call f injective or one-to-one if?

Answer 51

if, for all x1, x2 ∈ X, f (x1) = f (x2) implies x1 = x2.

Question 52

Let f : X → Y be a function. We call f surjective or onto if?

Answer 52

for each y ∈ Y there exists x ∈ X with f (x) = y.

Question 53

bijective function

Answer 53

f is both injective and surjective

Question 54

if f : X → Y is a bijection, then

Answer 54

there exists an inverse function f^−1 : Y → X such that (f ◦ f^−1)(y) = y for all y ∈ Y and (f −1 ◦ f )(x) = f (x) for all x ∈ X. Furthermore, f^−1 is also a bijection

Question 55

same cardinality

Answer 55

equal number of elements in both sets

Question 56

Two sets A and B are said to have the same cardinality if

Answer 56

there exists a bijection f : A → B. We write A ∼ B

Question 57

Proving A ∼ B

Answer 57

1. Define a function f : A → B. 2. Prove that f is injective. 3. Prove that f is surjective. 4. Conclude that f : A → B is a bijection and so A ∼ B.

Question 58

A ∼ B

Answer 58

A and B have the same cardinality (f every element of both A and B gets paired up, the sets have the same number of element)

Question 59

countable set

Answer 59

A ∼ N (natural numbers)

Question 59

sequence of set A

Answer 59

a function f : N → A where A = R (reals)

Question 60

uncountable set

Answer 60

neither finite nor countable

Question 61

finite set

Answer 61

there exists some n ∈ N (naturals) such that A ∼ {1, 2, . . . , n}

Question 62

term of a sequence

Answer 62

a sub n