Last updated: 4 November 2014
Sets, Functions, Relations, Posets,
elements, empty set, subset, union, intersection, disjoint, product of sets
injective, surjective, bijective, equal functions, inverse function, restriction, identity function, composition of functions.
(Important Theorem) Let be a function. An inverse function to exists if and only if is bijective.
finite, infinite, countable, uncountable.
HW: Show that
HW: Show that
Let be a set. A relation on is a subset of
Equivalence relation, partition of a set Equivalence class.
(Important Theorem) Let be a set.
(a) | Let be an equivalence relation on The set of equivalence classes of the relation is a partition of |
(b) | Let be a partition of The relation defined by if and are in the same is an equivalence relation on |
partially ordered set, totally ordered set, well ordered set.
upper/lower bound, maximal element, minimal element, smallest element, largest element.
Hasse diagram, lower/upper order ideal, intervals.
An ordered field is a field with a total order such that
(a) | If and then |
(b) | If and and then |
Let be an ordered field.
(a) | If and then |
(b) | If and then |
(c) | If and and then |
(d) |
If then |
(e) | If and and then if and only if |
(f) | |
(g) | If and and then |
HW: Show that with the usual order is an ordered field.
HW: Show that is a field and there does not exist an order on such that is an ordered field.
These are a typed copy of Lecture 47 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on October 20, 2014.