Real Analysis

Front 1

Set

Back 1

A collection of unique elements.

Front 2

Union

Back 2

A and B = {x in X: x in A and x in B}

Front 3

Intersection

Back 3

A or B = {x in X: x in A or x in B}

Front 4

Complement

Back 4

A^(c) = {x in X: x NOT in A)

Front 5

Set Difference

Back 5

A\B = A and B^(c) = {x in A and x not in B}

Front 6

Symmetric Difference

Back 6

A tri B = (A or B)\(A and B)

Front 7

Onto (surjective)

Back 7

Let X,Y be sets and f:X to Y. We say that f maps X onto Y if for each y in Y, there exists a point x in X such that f(x) =y.
in english, all the points in the domain must be hit.

Front 8

One to One (injective)

Back 8

Let X,Y be sets and f:X to Y. We say that f is one-to-one if each y in f(X) is mapped to by a unique element of X. Not all y in Y must be hit.

Front 9

Bijective

Back 9

both one to one and onto

Front 10

Domain

Back 10

the set the function is defined on

Front 11

Codomain

Back 11

all the values of the function.

Front 12

Range

Back 12

all the values in the Codomain which are mapped to.

Front 13

Well defined

Back 13

each point of x maps to exactly one point of Y.

Front 14

Cardinality

Back 14

Let A and B be sets. We say that A and B have the same cardinality, card(A) = card(B), if there is a bijection f: A to B.

Front 15

Cardinality of a finite set

Back 15

A set A is a finite set with cardinality n if card(A) = card(B) where B = {1,2,3,....,n}.

In other words, the cardinality of set A is the number of elements in

Front 16

Cardinality of the empty set

Back 16

0, no elements in the set

Front 17

Countable

Back 17

A set A is countable if card(A) = card(N)
or
A is finite

Front 18

Uncoutable

Back 18

otherwise

Front 19

Bounded above

Back 19

Let X be an ordered field and let A subset X be nonempty. We say A is bounded above if there is a point M in X such that a<=M for every a in A.

Front 20

Upper Bound

Back 20

M is the upper bound

Front 21

Bounded below

Back 21

A is bounded below if there is a point m in X such that m<=a for every a in A.

Front 22

Lower Bound

Back 22

Here m is the lower bound for A.

Front 23

Supremum (least upper bound)

Back 23

Let X be an ordered field and let A subset X be nonempty and bounded above. We say A has a supremum if there is a point in X, denoted sup(A), such that

a. sup(A) is an upper bound for A;
b. if b is another upper bound for A, then sup(A)<=b

Front 24

Complete

Back 24

We say that an ordered field X is complete if every set A subset X that is nonempty and bounded above has a least upper bound in X.

Front 25

Real number system

Back 25

We define the set R to be the complete ordered field containing Q as an ordered subfield.

Front 26

Absolute value function

Back 26

abs(x) = x, x>0
= 0, x = 0
= -x, x<0