Exam 1

Question 1

integer

Answer 1

Z = {…,-3,-2,-1,0,1,2,3,…}

Question 2

rational number

Answer 2

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

Question 3

irrational number

Answer 3

A number that can not be written as a rational number

Question 4

real number

Answer 4

An element of the set of real numbers

Question 5

the set of real numbers (p. 14)

Answer 5

An ordered field containing Q and satisfying the Axiom of Completeness

Question 6

upper bound

Answer 6

A set A ⊆ R is bounded above if there exists a number b∈R such that a≤b for all a∈A

Question 7

lower bound

Answer 7

A set A ⊆ R is bounded below if there exists a number l∈R such that l≤a for all a∈A

Question 8

least upper bound

Answer 8

i) s is an upper bound for A

ii) if b is any upper bound for A, then s≤b

Question 9

sup

Answer 9

least upper bound

Question 10

greatest lower bound

Answer 10

i) i is a lower bound for A

ii) if b is any lower bound for A, then b≤i

Question 11

inf

Answer 11

greatest lower bound

Question 12

max

Answer 12

b is a max of the set A if b∈A and b≥a for all a∈A

Question 13

min

Answer 13

b is a min of the set A if b∈A and b≤a for all a∈A

Question 14

bounded and unbounded
set

Answer 14

Has an upper and lower bound

Question 15

density

Answer 15

For every two numbers a and b with a<b, there exists a rational number r satisfying a<r<b.

Question 16

finite

Answer 16

(xi ∈ S, 1≤i≤n)

Question 17

countable

Answer 17

A set A is countable if N ~ A

Question 18

uncountable

Answer 18

Not countable

Question 19

cardinality

Answer 19

The size of a set. Number of elements

The set A has the same cardinality as B if there exists f : A → B that is 1-1 and onto. In this case, we write A ~ B

Question 20

one-to-one

Answer 20

if a1 ≠ a2, then f(a1) ≠ f(a2)

Question 21

onto

Answer 21

given any b∈B, it is possible to find an element a∈A for which f(a)=b

Question 22

power set

Answer 22

Given a set A, the power set P(A) refers to the collection of all subsets of A

Question 23

sequence

Answer 23

A sequence is a function whose domain is N

Question 24

convergence of a sequence (the ε − N definition),

Answer 24

A sequence (an) converges to a real number a if, for every positive number ε, there exists an N∈N such that whenever n≥N it follows that |an-a|< ε

Question 25

ε-neighborhood

Answer 25

Given a real number a∈R and a positive number ε>o the set Vsubε(a) = {x ∈ R: |x-a| < ε}

Question 26

divergent sequence

Answer 26

(∃L ∈ ℝ)(∀ 𝜖 > 0)(∃N ∈ ℕ)(∀n ∈ ℕ)[n ≥ N ⇒|𝑥𝑛−𝐿| < 𝜖]

Question 27

bounded sequence

Answer 27

A sequence (xn) is bounded if there exists a number M>0 such that |xn|≤ M for all n∈N

Question 28

Axiom of Completeness

Answer 28

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

Question 29

Nested Interval Property

Answer 29

For each n∈N assume we are given a a closed interval Isubn = [an,bn] = {x ∈ R : an≤x≤bn}. Assume also that each isubn contains Isub(n+1). Then, the resulting nested sequence of closed intervals

I1⊇I2⊇I3⊇I4⊇…
has a non-empty intersection

Question 30

Archimedean Property

Answer 30

i) Given any x∈R there exists an n∈N satisfying n>x
ii) Given any real number y>0, there exists n∈N satisfying1/n<y

Question 31

Algebraic limit theorem

Answer 31

Let lim an = a, and lim bn = b. Then,
(i) lim(can) = ca, for all c ∈ R;
(ii) lim(an + bn) = a + b;
(iii) lim(an*bn) = ab;
(iv) lim(an/bn) = a/b, provided b = 0.

Question 32

Monotone Convergence Theorem

Answer 32

If a sequence is monotone and bounded, then it converges.

A sequence is monotone if it is either increasing or decreasing. an≤a(n+1) or a(n+1) ≤ an