Real Analysis Definitions Flashcards

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 1-1 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
J-n
The set whose elements are the integers 1,2,..., n.
Finite set
A is finite if A~J-n 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.
k-cell
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 k-cell.
What is a 1-cell? a 2-cell?
An interval, a rectangle
Open (closed) ball B(x,r)
The set of all y in R^k such that |y-x| < (<=) r.
Convex set
We call a set E in R^k convex if
px + (1-p)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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Term:
Definition:
Definition:

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