Discrete Mathematics..

150 RelationsFigure 4.13. Let A = B = + , the set of positive integers, and let a R b denote ‘a hasthe same parity as b’; that is, either a and b are both even or they are bothodd. More preciselyR ={(a, b) : a − b is an integer multiple of 2}.Then 1 R 1, 1 R 3, 1 R 5,...2 R 2, 2 R 4, 2 R 6,...3 R 1, 3 R 3, 3 R 5,...4 R 2, 4 R 4, 4 R 6,...etc.A picture for this relation is figure 4.2, where again we have plotted theelements of R on the diagram for A × B.There are various ways of representing relations visually, particularly relationsbetween finite sets. In figures 4.1 and 4.2, the elements of R are marked on thecoordinate grid diagram of the Cartesian product A × B. Diagrams such as theseshow clearly R as a subset of A × B, but are not so good at showing additionalproperties of the relation.An alternative for finite sets is to represent A and B as two side-by-side Venndiagrams with the elements arranged vertically. An arrow is drawn from a ∈ Ato b ∈ B whenever a R b. We refer to this as the arrow diagram of the relation.For example, the arrow diagram for the relation defined in example 4.1.2 above isgiven in figure 4.3.Unfortunately figure 4.3 does not show very clearly at a glance which elementsare related to which. For sets larger than {1, 2, 3, 4, 5, 6} diagrams of this type