Mathematics for Computer Science

ly, taking a step amounts to applying a function, and going step by step corresponds to applying functions one after the other. This is captured by the operation of composing functions. Composing the functions f and g means that first f is applied to some argument, x, to produce f .x/, and then g is applied to that result to produce g.f .x//. Definition 4.3.1. For functions f W A ! B and g W B ! C , the composition, g ı f , of g with f is defined to be the function from A to C defined by the rule: .g ı f /.x/ WWD g.f .x//; for all x 2 A. Function composition is familiar as a basic concept from elementary calculus, and it plays an equally basic role in discrete mathematics. 4.4 Binary Relations Binary relations define relations between two objects. For example, “less-than” on the real numbers relates every real number, a, to a real number, b, precisely when a < b. Similarly, the subset relation relates a set, A, to another set, B , precisely when A B . A function f W A! B is a special case of binary relation in which an element a 2 A is related to an element b 2 B precisely when b D f .a/. In this section we’ll define some basic vocabulary and properties of binary relations. Definition 4.4.1. A binary relation, R, consists of a set, A, called the domain of R, a set, B , called the codomain of R, and a subset of A B called the graph of R. A relation whose domain is A and codomain is B is said to be “between A and B”, or “from A to B .” As with functions, we write R W A ! B to indicate that R is a relation from A to B . When the domain and codomain are the same set, A, we simply say the relation is “on A.” It’s common to use “a R b” to mean that the pair .a; b/ is in the graph of R.5 Notice that Definition 4.4.1 is exactly the same as the definition in Section 4.3 of a function, except that it doesn’t require the functional condition that, for each 5Writing the relation or operator symbol between its arguments is called infix notation. Infix expressions like “m < n” or “mC n” are the usual notation used for things like the less-then relation or the addition operation rather than prefix notation like “< .m; n/” or “C.m; n/.” “mcs” — 2015/5/18 — 1:43 — page 90 — #98 Chapter 4 Mathematical Data Types 90 domain element, a, there is at most one pair in the graph whose first coordinate is a. As we said, a function is a special case of a binary relation. The “in-charge of” relation, Chrg, for MIT in Spring ’10 subjects and instructors is a handy example of a binary relation. Its domain, Fac, is the names of all the MIT faculty and instructional staff, and its codomain is the set, SubNums, of subject numbers in the Fall ’09–Spring ’10 MIT subject listing. The graph of Chrg contains precisely the pairs of the form .hinstructor-namei ; hsubject-numi/ such that the faculty member named hinstructor-namei is in charge of the subject with number hsubject-numi that was offered in Spring ’10. So graph.Chrg/ contains pairs like .T. Eng; 6.UAT/ .G. Freeman; 6.011/ .G. Freeman; 6.UAT/ .G. Freeman; 6.881/ .G. Freeman; 6.882/ .J. Guttag; 6.00/ .A. R. Meyer; 6.042/ .A. R. Meyer; 18.062/ .A. R. Meyer; 6.844/ .T. Leighton; 6.042/ .T. Leighton; 18.062/ :: (4.4) Some subjects in the codomain, SubNums, do not appear among this list of pairs—that is, they are not in range.Chrg/. These are the Fall term-only subjects. Similarly, there are instructors in the domain, Fac, who do not appear in the list because they are not in charge of any Spring term subjects. 4.4.1 Relation Diagrams Some standard properties of a relation can be visualized in terms of a diagram. The diagram for a binary relation, R, has points corresponding to the elements of the domain appearing in one column (a very long column if domain.R/ is infinite). All the elements of the codomain appear in another column which we’ll usually picture as being to the right of the domain column. There is an arrow going from a point, a, in the lefthand, domain column to a point, b, in the righthand, codomain column, precisely when the corresponding elements are related by R. For example, here are diagrams for two functions: “mcs” — 2015/5/18 — 1:43 — page 91 — #99 4.4. Binary Relations 91 A B a 1 b PPPPPq 2 c PPPPPq 3 d 3 4 e 3 A B a 1 b PPPPPq 2 c Q Q Q Q QQs 3 d 3