By Seymour Lipschutz
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Sample text
Thus Rn is de®ned for all positive n. Warning: Many texts denote the composition of relations R and S by S R rather than R S. This is done in order to conform with the usual use of g f to denote the composition of f and g where f and g are functions. Thus the reader may have to adjust his notation when using this text as a supplement with another text. However, when a relation R is composed with itself, then the meaning of R R is unambiguous. The arrow diagrams of relations give us a geometrical interpretation of the composition R S as seen in the following example.
A) Find the matrix MR of R. (d ) (b) Find the domain and range of R. (e) (c) Find RÀ1 : Draw the directed graph of R. 31. R S; and (b) S f b; a; c; c; c; d; d; ag S R; (c) R R; (d ) S S: Let R be the relation on the positive integers N de®ned by the equation x 3y 12; that is, R f x; y: x 3y 12g (c) V Â V. Lipschutz−Lipson: Schaum’s Outline of Theory and Problems of Discrete Math, 2/e 2. Relations © The McGraw−Hill Companies, 2004 Text CHAP. 2] 47 RELATIONS (a) Write R as a set of ordered pairs.
35. Prove that if R is an equivalence relation on a set A, then RÀ1 is also an equivalence relation on A. 36. Let S f1; 2; 3; F F F ; 19; 20g. Let R be the equivalence relation on S de®ned by x y (mod 5), that is, x À y is divisible by 5. 37. Let A f1; 2; 3; F F F ; 9g and let $ be the relation on A Â A de®ned by a; b $ c; d if (a) Prove that $ is an equivalence relation. 25. (a) (b) (c) W Â V f(Marc, Erik), (Marc, David), (Erik, Erik), (Erik, David), (Paul, Erik), (Paul, David)}. V Â W f(Erik, Marc), (David, Marc), (Erik, Erik), (David, Erik), (Erik, Paul), (David, Paul)}.