# 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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