Discrete Structure lecture_11

Apr 28th, 2015
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It includes all the details about the Discrete structure with examples.Step by step solution.

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DISCREAT STRUCTURELECTURE # 11INVERSE OFRELATIONSKHAWAJA TARIQ MAHMOODLecture # 11INVERSE OF RELATIONSINVERSE OF RELATION:Let R be a relation from A to B. The inverse relation R-1 from B to A is defined as:R-1 = {(b,a) BA | (a,b) R}More simply, the inverse relation R-1 of R is obtained by interchanging theelements of all the ordered pairs in R.EXAMPLE:Let A = {2, 3, 4} and B = {2, 6, 8} and let R be the divides relation from A to Bi.e. for all (a,b) A B, a R b a | b (a divides b) Then R = {(2,2), (2,6), (2,8), (3,6), 4,8)}andR-1= {(2,2), (6,2), (8,2), (6,3), (8,4)}In words, R-1may be defined as:for all (b,a) B A, b R a b is a multiple of a.ARROW DIAGRAM OF AN INVERSE RELATION:The relation R = {(2,2), (2,6), (2,8), (3,6), (4,8)} is represented by the arrow diagram.234A268BThen inverse of the above relation can be obtained simply changing the directions of the arrowsand hence the diagram isKhawaja Tariq Mahmood.Page 2Lecture # 11INVERSE OF RELATIONSR-1A234268BMATRIX REPRESENTATION OF INVERSE RELATION:The relation R = {(2, 2), (2, 6), (2, 8), (3, 6), (4, 8)} from A = {2, 3, 4} toB = {2, 6, 8} is defined by the matrix M below:2 6 82 3 42 1 1 1M 3 0 1 04 0 0 12 1 0 0 M t 6 1 1 0 8 1 0 1 The matrix representation of inverse relation R-1 is obtained by simply taking itstranspose. Hence R-1 is repre

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