Foundation of Mathematics - UQU

Principle of Mathematical Logic: Statements, negation of statement, Connectives and truth tables, Methods of proof (direct proof, proof of the contrapositive, proof by contradiction, proof by counter-example, proof by mathematical induction) 2 Sets and algebra of sets: Method of defining a set (listing method, characteristic property), finite and infinite sets. Membership and inclusion, Universal and existential quantifiers. Power set, Algebra of sets (union, intersection, universal set, complement of a set, symmetric difference, De Morgan’s laws, Venn diagrams, repetition, sets of numbers (N, Z, Q, R and C) 3 Cartesian Product and Relations on Sets: Cartesian product of sets, ordered pairs, Binary relations on sets, reflexive, symmetric, transitive relations, Skew-symmetric (Anti-symmetric), Equivalence relation, ordered relation, Partition of sets and equivalence classes, Partial ordered relation, Inverse of relation, Composition of relations. Diagrams of relations 4 Mappings: Definition of mapping, Image of mapping, Inverse image of mapping Special types of mappings (injective (1-1), surjective (onto), bijective (1-1 and onto), Identity mapping, Composition of mappings, Bijection mappings as permutations, inverse of mapping. Equivalence of sets 5 Binary Operations: Definition and examples of binary operations, closure of a binary operation, commutative and associative operations, Identity element, Inverse of element, Systems of two operations, Homomorphism between two closed algebraic systems 6 Introduction to Groups: Semigroups (definition and examples), Groups (definition and examples), Subgroups (definition and examples), examples of groups and subgroups 7 Introduction to Rings and Fields: Definition and examples of rings. Definition and examples of fields, some properties of rings and fields. Ring of polynomials