Discrete Mathematics: Chapter 7, Posets, Lattices, & Boolean Algebra

Algebra deals with more than computations such as addition or exponentiation; it also studies relations. Calculus touches on this a bit with locating extreme values and determining where functions increase and decrease; and in elementary algebra you occasionally “solve” inequalities involving the order relations of < or ≤ , but this almost seems like an intrusion foreign to the main focus, which is making algebraic calculations. Relational ideas have become more important with the advent of computer science and the rise of discrete mathematics, however. Many contemporary mathematical applications involve binary or n-ary relations in addition to computations. We began discussing this topic in the last chapter when we introduced equivalence relations. In this chapter we will explore other kinds of relations (these will all be binary relations here), particularly ones that impose an order of one sort or another on a set. This will lead us to investigate certain order-structures (posets, lattices) and to introduce an abstract type of algebra known as Boolean Algebra. Our exploration of these ideas will nicely tie together some earlier ideas in logic and set theory as well as lead us into areas that are of crucial importance to computer science.