Discrete Mathematics for New Technology Book

Logic: Propositions and truth values. Logical connectives and truth tables. Tautologies and contradictions. Logical equivalence and logical implication. The algebra of propositions. More about conditionals. Arguments. Predicate logic. Arguments in predicate logic. Mathematical proof: The nature of proof. Axioms and axiom systems. Methods of proof. Mathematical induction. Sets: Sets and membership. Subsets. Operations on sets. Counting techniques. The algebra of sets. Families of sets. The cartesian product. Relations: Relations and their representations. Properties of relations. Intersections and unions of relations. Equivalence relations and partitions. Order relations. Hasse diagrams. Application: relational databases. Functions: Definitions and examples. Composite functions. Injections and surjections. Bijections and inverse functions. More on cardinality. Databases: functional dependence and normal forms. Matrix algebra: Introduction. Some special matrices. Operations on matrices. Elementary matrices. The inverse of a matrix. Systems of linear equations: Introduction. Matrix inverse method. Gauss-Jordan elimination. Gaussian elimination. Algebraic structures: Binary operations and their properties. Algebraic structures. More about groups. Some families of groups. Substructures. Morphisms. Group codes. Boolean algebra: Introduction. Properties of Boolean algebras. Boolean functions. Switching circuits. Logic networks. Minimization of Boolean expressions. Graph theory: Definitions and examples. Paths and circuits. Isomorphism of graphs. Trees. Planar graphs. Directed graphs. Applications of graph theory: Introduction. Rooted trees. Sorting. Searching strategies. Weighted graphs. The shortest path and travelling salesman problems. Networks and flows. References and further reading. Hints and solutions to selected exercises. Index.