Schaum's Discrete Mathematics by Seymour Lipschutz

By Seymour Lipschutz

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Example text

Then …a; b† and …b; c† belong to S for every S. Since each S is transitive, …a; c† belongs to S for every S. Hence, …a; c† P T and T is transitive. 16. Let R be a relation on a set A, and let P be a property of relations, such as, symmetry and transitivity. Then P will be called R-closable if P satis®es the following two conditions: (1) There is a P-relation S containing R. (2) The intersection of P-relations is a P-relation. (a) Show that symmetry and transitivity are R-closable for any relation R.

4: Rà is the transitive closure of a relation R. Suppose A is a ®nite set with n elements. Then we show in Chapter 8 on directed graphs that Rà ˆ R ‘ R2 ‘ Á Á Á ‘ Rn This gives us the following result. 5: Let R be a relation on a set A with n elements. Then transitive…R† ˆ R ‘ R2 ‘ Á Á Á ‘ Rn Finding transitive…R† can take a lot of time when A has a large number of elements. An ecient way for doing this will be described in Chapter 8. Here we give a simple example where A has only three elements.

R3 is not symmetric since …1; 3† P R3 but …3; 1† P = R3 . The R1 is not symmetric since …1; 2† P R1 but …2; 1† P other relations are symmetric. 7 are symmetric. The relation c is symmetric since if line a is perpendicular to line b then b is perpendicular to a. Also, k is symmetric since if line a is parallel to line b then b is parallel to a. The other relations are not symmetric. For example, 3 4 but 4M3; f1; 2g  f1; 2; 3g but f1; 2; 3g \ f1; 2g, and 2j6 but 6j=2. A relation R on a set A is antisymmetric if whenever aRb and bRa then a ˆ b, that is, if whenever …a; b†, …b; a† P R then a ˆ b.

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