Class Notes Review of elementary set theory :Algebra of sets – Ordered pairs and Cartesian products –Countable and Uncountable sets Relations :-Relations on sets –Types of relations and their properties – Relational matrix and the graph of a relation – Partitions – Equivalence relations - Partial ordering- Posets – Hasse diagrams - Meet and Join – Infimum and Supremum Functions :- Injective, Surjective and Bijective functions - Inverse of a function- Composition Module 1 Review of Permutations and combinations, Principle of inclusion exclusion, Pigeon Hole Principle, Recurrence Relations: Introduction- Linear recurrence relations with constant coefficients– Homogeneous solutions – Particular solutions – Total solutions Algebraic systems:- Semigroups and monoids - Homomorphism, Subsemigroups and submonoids Module 2 Algebraic systems (contd...):- Groups, definition and elementary properties, subgroups, Homomorphism and somorphism, Generators - Cyclic Groups, Cosets and Lagrange’s Theorem Algebraic systems with two binary operations- rings, fields-sub rings, ring homomorphism Module 3 Lattices and Boolean algebra :- Lattices –Sublattices – Complete lattices – Bounded Lattices - Complemented Lattices – Distributive Lattices – Lattice Homomorphisms. Boolean algebra – sub algebra, direct product and homomorphisms Module 4 Predicate Logic:- Predicates – Variables – Free and bound variables – Universal and Existential Quantifiers – Universe of discourse. Logical equivalences and implications for quantified statements – Theory of inference : Validity of arguments. Proof techniques: Mathematical induction and its variants – Proof by Contradiction – Proof by Counter Example – Proof by Contra positive. Links Product of Poset - (Similar diagrams are obtained for Lattice and Boolean Algebra) Abstract Algebra: Theory and Applications - (Book with Sage examples) Assignment Questions [Series 2 ] [Answer Key] |

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