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Representing Relations Rosen 7.3

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Using Matrices For finite sets we can use zero-one matrices. Elements of each set A and B must be listed in some particular (but arbitrary) order. When A=B we use the same ordering for A and B. m ij = 1 if (a i,b j ) R = 0 if (a i,b j ) R

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Example Zero-One Matrix b1b2b3 a1 a2 a3 R = {(a1,b1), (a1,b2), (a2,b2), (a3,b2), (a3,b3)}

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Matrix of a relation on a set, A Can be used to determine whether the relations has certain properties. Recall that R on A is reflexive if (a,a) R for every element a A. ReflexiveNot Reflexive

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A relation R on a set A is called Symmetric if (b,a) R whenever (a,b) R for a,b A. M R = (M R ) t is Antisymmetric if (a,b) R and (b,a) R only if a=b for a,b A is antisymmetric. –If m ij = 1, i j, m ji = 0 SymmetricAntisymmetricNeither

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Examples Reflexive Symmetric Reflexive Antisymmetric

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Let R1, R2 be relations on A A = {1,2,3} R1 = {(1,1), (1,3), (2,1), (3,3)} R2 = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,1)}

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R1 R2, R1 R2 M R1 R2 = M R1 M R2, M R1 R2 = M R1 M R2

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What is R2 R1? The composite of R 1 and R 2 is the relation consisting of ordered pairs (a,c) where a A, c A, and for which there exists an element b A such that (a,b) R 1 and (b,c) R 2. R2 R1 = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1)} R1 = {(1,1), (1,3), (2,1), (3,3)} R2 = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,1)}

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Boolean Product Let A = [a ij ] be an m by k zero-one matrix and B = [b ij ] be a k by n zero-one matrix. Then the Boolean Product of A and B denoted by A B is the m by n matrix with i,j entry c ij where c ij = (a i1 b 1j ) (a i2 b 2j ) ... (a ik b kj ).

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What is R2 R1? R2 R1 = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1)} M R2 R1 =M R1 M R2

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Directed Graphs (Digraph) A directed graph consists of a set V of vertices together with a set E of ordered pairs of elements of V called edges. –(a,b), a is initial vertex, b is the terminal vertex a b c Reflexive (Loops at all vertices) Symmetric (All edges both ways)

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Relation R on a set A a b c R = {(a,b), (b,b), (b,c), (c,a), (c,c)} Transitive? No a b c R = {(a,b), (b,b), (b,c), (a,c), (c,c)} Transitive? Yes Rosen, pp. 493-494

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Relation R on a set A a b c R = {(a,a), (a,c), (b,b), (b,a), (b,c), (c,c)} Reflexive Antisymmetric Transitive

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