Adv. Calculus Final

Front 1

Order

Back 1

An order on S is a relation, denoted by <, with the following two properties:

i) If x ϵ S and y ϵ S then one and only one of these statements is true: x < y, x = y, x > y

ii) If x, y, z ϵ S, if x< y and y < z then x < z

Front 2

Bounded Above/ Upperbound

Back 2

Consider E C S, an ordered set. E is bounded above, if there exists a B ϵ S s.t.

x <= B for all x ϵ E

In this case, B is called an upperbound of E. (same for alpha lowerbound)

Front 3

Least Upper Bound/ Supremum of E.

Back 3

Suppose E C S, an ordered set, and E is bounded above. Suppose a exists w/ these properties:

i) a is an upperbound of E.

ii) y < a => y is not an upper bound of E.

Then a = the least upper bound of E.

a = sup E

(same for a = greatest lower bound of E/ inf E)

Front 4

Least-Upper-Bound Property (LUBP)

Back 4

An ordered set S has the LUBP if

- empty set(0/) does not equal E C S, E bounded above => sup E exists in S.

Front 5

Field

Back 5

A field is a set F w/ 2 operations (+, mult.) which satisfy the "field axioms" x, y , z ϵ F

Addition: Multiplication:

(A1) x + y ϵ F (M1) xy ϵ F

(A2) x + y = y + x (M2) xy = yx

(A3) x + (y + z) = (x+y)+z (M3) (xy)z = x(yz)

(A4) 0 ϵ F s.t. 0+x=X (M4) 1ϵF s.t. 1x=x

(A5) -xϵF s.t. (-x)+x=0 (M5) 1/xϵF s.t. 1/x*x=1

Front 6

Ordered Field

Back 6

An ordered field is a field F which also is an ordered set and x, y, z ϵF

i) y < z => x +y < x+z

ii) x > 0, y > 0 => xy > 0

Front 7

Complex conjugate z̅

Back 7

If z = a + ib then z bar = a - ib

Front 8

Image

Back 8

Let A and B be two sets and f be a mapping of A into B. If E C A, f(E) is defined to be the set of all elements f(x), for x ϵ E. We call f(E) the image of E under f. f(A) is the range

Front 9

Onto

Back 9

If f(A) = B, f maps A onto B.

f ^-1(E) is the inverse image of E.

Front 10

1-1

Back 10

If for each x ϵ B, f^-1(y) consists of <= 1 element of A, then f is a 1-1 mapping provided that f(x1) does not equal f(x2) when x1 does not equal x2

Front 11

1-1 correspondence equivalent

Back 11

If there is a 1-1 mapping of A onto B, A and B can be put into 1-1 correspondence or that card(A) = card(B) or A,B are equivalent.

Card = # of elements in a set

Front 12

Finite

Infinite

Countable

Uncountable

At most countable

Back 12

For n ϵ natural N > 0, let N(n) := {1,2,...,n} for any set A:

a) A is finite if A ~ N(n) for some n ϵ natural N

b) A is infinite if A is not finite

c) A is countable if A ~ N(n)

d) A is uncountable if A is neither finite nor countable

e) A is at most countable if A is finite or count.

Front 13

Sequence

Back 13

A sequence {Xn} C A is a function w/ domain natural N (f(n) = Xn)

Front 14

Metric space

Back 14

Set X is a metric space if for all p, q ϵ X there is some d(p, q) ϵ Reals :

a) d(p,q) > 0 if p does not equal q

b) d(p,q) = d(q,p)

c) d(p,q) <= d(p,r) + d(r,q) for all r ϵ X

Front 15

Neighborhood

Back 15

A neighborhood of a point p is a set Nr(p) consisting of all points q s.t. d(p,q) < r. r is the radius of Nr(p)

Front 16

Limit Point

Back 16

A point p is a limit point of the set E if every neighborhood of p contains a point q not equal to p s.t. q ϵ E

Front 17

Isolated Point

Back 17

If p ϵ E and p is not a limit point of E, then p is called an isolated point of E

Front 18

Closed

Interior

Open

Complement

Back 18

E is closed if every limit point of E is a point of E.

A point p is an interior point of E if there is some Nr(p) s.t. N C E.

E is open if every point of E is an interior point of E.

The complement of E(denoted E ^ c) is the set of all points p ϵ X s.t. p is not ϵ E

Front 19

Perfect

Bounded

Dense

Back 19

E is perfect if E is closed and if every point of E is a limit point of E.

E is bounded if there is a real number M and a point q ϵ X s.t. d(p, q) < M for all p ϵ E.

E is dense in X if every point of X is a limit point of E, or a point of E( or both)

Front 20

closure

Back 20

Let E c X, a metric space. Let E':={limit points of E}

Then the closure of E is

E bar = close(E) E U E'

Front 21

Open Cover

Back 21

Let E be a set contained in a metric space X. By open cover we mean a collection { G a} of open subsets of X s.t. E c UaGa

Front 22

Compact

Back 22

A subset K of a metric space X is compact if every open cover contains a finite subcover.

Front 23

Converges (diverges)

Back 23

A sequence {pn} c X, a metric space, converges if for p ϵ X all E >0 there is some N ϵ natural N :

n >= N => d(pn, p) < E

(p = lim as n->oo pn or pn -> p. If a sequence does not converge, it diverges.)

Front 24

Subsequence

Subsequential Limit

Back 24

A subsequence of {pn} is obtained from taking terms {pnk} where n1 < n2 < n3<...

If {pnk} converges, then its limit is a subsequential limit

Front 25

Cauchy Sequence

Back 25

{pn} c X metric is a Cauchy sequence if

for all E > 0 there is some N :

n,m >= N => d(pn, pm) < E

Front 26

Diameter of E

Back 26

empty set does not = E c X metric and let

s:= {d(p,q) : p,q ϵ E}.

Diam E:= sup S

Front 27

Cpmplete

Back 27

A metric sequence is complete if every Cauchy sequence converges

Front 28

Upper Limit

Lower Limit

Back 28

Let {sn} ϵ R and there be the set of x s.t. snk -> x for a subsequence {snk}.

s* = sup E

slow * = inf E

s*, sLow* are called the upper and lower limits of {sn};

lim n->oo sup sn = s*, lim n->oo inf sn =slow*

Front 29

Infinite Series/Series

Back 29

The summation from n = 1 to oo of An = S

if Sn = the summation from k = 1 to n of Ak is the partial series.

Front 30

Power Series (coefficients

Back 30

Fix a sequence {Cn} c Complex numbers

The summation from n = 0 to oo of Cnz^n, z ϵ Complex

is a power series

(Cn are the coefficients of the series)

Front 31

Product of the summation of An and Bn

Back 31

Given summation An, summation Bn, then

Cn = summation from k = 0 to n of AkBn-k, n ϵ N0 is the product of the two summations.

Front 32

X, Y metric spaces, f: Ec X -> y, p limit point of E

f(x) -> q as x -> p or lim x->p f(x) = q if...

Back 32

there is q ϵ Y s.t.

for all E>0 there is some delta> 0 : 0< dX(x,p) < delta