### 2019–20–B

## Dr. Stewart Smith

## Course topics

Part A: Logic and set-theory. Propositional calculus, Boolean operations. Truth tables, the truth-value of a propositional formula (without induction at this stage), logical implication and logical equivalence, tautologies and contradictions, the useful tautologies, distributivity and de-Morgan’s Law. Sets: the notion of a set, membership and equality, operations: union, intersection, set-difference and power-set. Ordered pairs and Cartesian products. Equivalence relations, quotient spaces and partitions.Partial orders. Functions, injective and surjective functions, invertibility of a function. The ordered set of natural numbers.The axiom of induction in different forms.

Part B: Finite and infinite sets. The notion of cardinality. Countable sets. Cantor’s theorem on the power set of a set.

Part C: Combinatorics. Basic counting formulas. Binomials. Inclusion-exclusion technique. Recursive definition and formulas.

Part D: Graph Theory. Graphs, examples, basic facts, vertex degrees, representing a graph, neighborhood matrices, connected components, Euler graphs, bipartite graphs, matching in bipartite graphs, Hall’s marriage theorem, graph colorings.