Chapter 1 (part b)

Question 1

Nested Interval Property

For each n in N, assume we are given a closed interval

I_n = [a_n,b_n] = {x in R: a_n < x < b_n}. Assume also that each In

contains I_n+1. Then….

Answer 1

the resulting nested sequence of closed intervals

… c I₃ c I₂ c I₁

has a non-empty intersection.

Or Π_{n=1 to inf} (I_n) ≠ 0.

Question 2

Archimedean Property

i. ) Given any number x in R, …
ii. ) Given any real number y > 0, ….

Answer 2

i. ) there exists an n in N such that n > x.
ii. ) there exists an n in N satisfying 1/n

Question 3

Cut Property

If A and B are non-empty disjoint sets with

AUB = R and a < b for all a in A and b in B

Answer 3

then there exists c in R such that x < c whenever x in A and x > c whenever x in B.

Question 4

Density of Q in R

For every two real numbers a and b with a

Answer 4

there exists a rational number r satisfying

a < r < b

Question 5

Density of I in R

Given any two real numbers a and b, where a < b…

Answer 5

there exists a t not in Q such that

a < t < b

Question 6

Cardinality

Answer 6

the size of a set

Question 7

Same Cardinality

Two sets have the same cardinality if…

Answer 7

there exists a one-to-one correspondence between elements of a set and the function is onto.

A~B

[A is equivalent to B]

Question 8

Bijective

Answer 8

a function f that is one-to-one and onto

Question 9

Countable

a set A is countable if…

Answer 9

N~A

N has the same cardinality as A

[N is equivlanent to A]

Question 10

Theorem- If A c B and B is countable…

Answer 10

then A is either finite or countable.

Question 11

the resulting nested sequence of closed intervals

… c I₃ c I₂ c I₁

has a non-empty intersection.

Or Π_{n=1 to inf} (I_n) ≠ 0.

Answer 11

Nested Interval Property

For each n in N, assume we are given a closed interval

I_n = [a_n,b_n] = {x in R: a_n < x < b_n}. Assume also that each In

contains I_n+1. Then….

Question 12

i. ) there exists an n in N such that n > x.
ii. ) there exists an n in N satisfying 1/n

Answer 12

Archimedean Property

i. ) Given any number x in R, …
ii. ) Given any real number y > 0, ….

Question 13

then there exists c in R such that x < c whenever x in A and x > c whenever x in B.

Answer 13

Cut Property

If A and B are non-empty disjoint sets with

AUB = R and a < b for all a in A and b in B…

Question 14

there exists a rational number r satisfying

a < r < b

Answer 14

Density of Q in R

For every two real numbers a and b with a < b…

Question 15

there exists a t not in Q such that

a < t < b

Answer 15

Density of I in R

Given any two real numbers a and b, where a < b…

Question 16

the size of a set

Answer 16

Cardinality

Question 17

there exists a one-to-one correspondence between elements of a set and the function is onto.

A~B

[A is equivalent to B]

Answer 17

Same Cardinality

Two sets have the same cardinality if…

Question 18

a function f that is one-to-one and onto

Answer 18

Bijective

Question 19

N~A

N has the same cardinality as A

[N is equivlanent to A]

Answer 19

Countable

a set A is countable if…

Question 20

then A is either finite or countable.

Answer 20

Theorem- If A c B and B is countable…

Question 21

Theorem- The open interval (0,1)

={x in R| 0 < x < 1}

is…

Answer 21

uncountable.

Question 22

Given a set A, the power set P(A) refers to…

Answer 22

the collection of all subsets of A.

Question 23

the collection of all subsets of A.

Answer 23

Given a set A, the power set P(A) refers to…

Question 24

Cantor’s Theorem

Given any set A, there does not exist a function…

Answer 24

f:A to P(A) that is onto.

(The power set of N is uncountable)

Question 25

f:A to P(A) that is onto.

(The power set of N is uncountable)

Answer 25

Cantor’s Theorem

Given any set A, there does not exist a function…

Question 26

The sets A and B appear in the same equivalence class

if and only if

Answer 26

A and B have the same cardinality.

Question 27

A and B have the same cardinality.

Answer 27

The sets A and B appear in the same equivalence class

if and only if

Question 28

Schroder-Bernstein Theorem

Assume there exists a one to one function

f: X to Y and another one to one function
g: Y to X, then…

Answer 28

there exists a one to one and onto function

h: X to Y and therefore

X ~ Y.

Question 29

there exists a one to one and onto function

h: X to Y and therefore

X ~ Y.

Answer 29

Schroder-Bernstein Theorem

Assume there exists a one to one function

f: X to Y and another one to one function
g: Y to X, then…

Question 30

A real number x in R is called algebraic if

there exists integers a₀, a₁, a₂…

not all zero such that…

Answer 30

a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ = 0

(if it is the root of a polynomial with

integer coefficients)

Question 31

a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ = 0

(if it is the root of a polynomial with

integer coefficients)

Answer 31

A real number x in R is called algebraic if

there exists integers a₀, a₁, a₂…

not all zero such that…

Question 32

Theorem-

i. ) If A₁, A₂, …A_m are countable…
ii. ) If A_n is countable for all n in N, then…

Answer 32

i. ) Ո_{n=1 to inf} A_n is also countable.
ii. ) U_{n=1 to inf} A_n is also countable.

Question 33

i. ) Ո_{n=1 to inf} A_n is also countable.
ii. ) U_{n=1 to inf} A_n is also countable.

Answer 33

Theorem-

i. ) If A₁, A₂, …A_m are countable…
ii. ) If A_n is countable for all n in N, then…