Advanced Set Theory and Lattices: Countable and uncountable sets, The concept of cardinal numbers. Addition and multiplication of cardinals. Partially and Totally ordered sets. Well-ordered sets. Ordinals numbers. Ordinal addition and multiplication. Cartesian products and choice function, axiom of choice, statement of Zorn’s Lemma. Diagramatical representation of a poset. Isomorphism, Duality, Product of two posets. Semi Lattices. SubLattices.
Course Learning Outcomes
Upon completion of this course, the student will be able to understand basic properties of ordered sets; their diagrams; maps between ordered sets; the duality principle; down-sets and up-sets; maximal and minimal elements; top and bottom; and building
new ordered sets,, Learning lattices as ordered sets; complete lattices; chain conditions and completeness;
and how to construct complete partially ordered sets.
Provides opportunity of learning further rich topics of set theory like the existence of maximal elements and the celebrated Zorn’s Lemma.
This course further enables students how to deal with lattices as algebraic structures; to form sublattices; products; homomorphisms.
Lattice theory, although a comparatively new branch of Mathematics can no more be treated as still in its infancy. It is fast making inroads into diverse mathematical disciplines and has already made its presence felt in the fields of Toplogy, analysis, algebra, Geometry, Probability and logic.
Countable and uncountable sets
unit interval is uncountable
Book Title : Abstract Set Theory
Author : A.A. Fraenkal
Edition : THIRD EDITION
Publisher : North-Holland Publishing Company
Book Title : Axiomatic Set Theory
Author : P. Suppes
Edition : SECOND EDITION
Publisher : Dover Publication
Book Title : Naïve Set Theory
Author : P.R. Holmos
Book Title : Theory and Problem of Set theory and related Problems
Author : Seymour Lipschutz
Publisher : McGraw-Hill Publishing company