Terms  Definitions 

Function 
A mapping from set A into set B.

Domain 
The set the function is defined on.

Range 
The set of all values of a function.

Image 
Let E be a subset of the domain. Then f(E) is the image of E under f.

Onto function 
If f(A) = B then f is onto.

1 to 1 function 
If for each element y of B, the inverse value for y in A is at most one element of A, then f is a 11 function of A into B.

Inverse image 
If E is a subset of B, then f(1)(E) is the set of all x in A such that f(x) is in E.

Requirements for set equivalence 
Reflexive, symmetric and transitive

Reflexive 
A ~ A

Symmetric 
If A ~ B then B ~ A

Transitive 
If A~B and B~C then A~C

Jn 
The set whose elements are the integers 1,2,..., n.

Finite set 
A is finite if A~Jn for some n (the empty set is also finite)

Infinite set 
A set which is not finite

Countable set 
A is countable if A~J

Uncountable set 
If A is neither finite nor countable, then it is uncountable.

At most countable set 
A is at most countable if A is finite or countable.

T/F: A finite set may be equivalent to one of its proper subsets. 
False. This may be true for infinite sets, though.

Sequence 
A function defined on J.

Theorem: Every infinite subset of a countable set A is countable. 
Proof:

T/F: The set of all rationals is countable. 
True.

Theorem: Let A be the set of all sequences whose elements are the digits 0 and 1. This set A is uncountable. 
Proof: Cantor's diagonal process.

Metric Space 
Let X be a set with elements called points. If p and q are in X, then there is a distance function d(p,q) which is metric.

Three characteristics of a metric. 
1. d(p,q) > 0 if p not q, = 0 if p = q.
2. d(p,q) = d(q,p) 3. d(p,q) <= d(p,r) + d(r,q) for any r in X. 
T/F: Every subset of a metric space is a metric space. 
True.

Segment (a,b) 
The set of all real numbers x such that a < x < b.

Interval [a,b] 
The set of all real numbers x such that a <= x <= b.

kcell 
If a(i) < b(i) for i=1,..,k, the set of all points x=(x(1),x(2),...,x(k)) in R(K) whose coordinates satisfy the inequalities a(i) <= x(i) <= b(i), i=1,...,k is called a kcell.

What is a 1cell? a 2cell? 
An interval, a rectangle

Open (closed) ball B(x,r) 
The set of all y in R^k such that yx < (<=) r.

Convex set 
We call a set E in R^k convex if
px + (1p)y is in E whenever x,y in E and 0 < p < 1. 
T/F: Balls are convex 
T

Neighborhood 
Neighborhood of a point p is a set Nr(p) consisting of all pokints q such that d(p,q) < r.
r is called the radius of Nr(p) 
Limit point of E 
A point p is a limit point of a set E if every nbhd of p contains a point q <> p such that q is in E.

Isolated point of E 
A member of E which is not a limit point.

Closed set 
One which contains all its limit points.

Interior point of E 
A point p is an interior point of E if there is a nbhd N of p such that N is in E.

Open set 
One where all its members are interior points.

Complement of a set E 
The set of all points in the space not in E.

Perfect set E 
E is closed and all its members are limit points.

A bounded set E 
E is bounded if there is a real number M and a point q in X such that d(p,q) < M for all p in E.

E is dense in X means 
Every point of X is a limit point of E, or a point of E, or both.

Thm: Every neighborhood is an open set. 
Proof from definitions.

Thm. If p is a limit point of E, then every nbhd of p contains infinitely many points of E. 
Proof by contradiction.

Cor. A finite point set has no limit points. 
Proof.

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