Sets

This chapter introduces the fundamental notation, concepts and operations for reasoning with sets and applies these in a number of modelling examples. The first section presents set builder notation to declare the set of all elements with a certain property; introduces the notion of cardinality of a finite set; and introduces notation for common sets in mathematics such as the empty set and the set of natural numbers. The next section discusses the notion of set membership and defines what it means for two sets to be equal, and for one set to be a subset of another set. The correspondence between sets and properties is further explored in the next two sections, including a discussion of Russell’s paradox. Next, the standard operations of union, intersection, difference and complementation are defined and more advanced properties such as power sets, generalised unions and intersections, as well as ordered pairs, tuples and Cartesian products are introduced. Modelling and reasoning with sets is practiced in a number of puzzles and examples. Algebraic laws for reasoning with sets, essentially those of Boolean algebras, are explored in the final section.