Composition of RelationsLet A, B, and C be sets, and let R be a relation from A to B and let S be a relation from B to C. That is, R is a subset of A × B and S is a subset of B × C. Then R and S give rise to a relation from A to C indicated by R◦S and defined by: The relation R◦S is known the composition of R and S; it is sometimes denoted simply by RS. Let R is a relation on a set A, that is, R is a relation from a set A to itself. Then R◦R, the composition of R with itself, is always represented. Also, R◦R is sometimes denoted by R^{2}. Similarly, R^{3} = R^{2}◦R = R◦R◦R, and so on. Thus R^{n} is defined for all positive n. Example1: Let X = {4, 5, 6}, Y = {a, b, c} and Z = {l, m, n}. Consider the relation R_{1} from X to Y and R_{2} from Y to Z. R_{1} = {(4, a), (4, b), (5, c), (6, a), (6, c)} R_{2} = {(a, l), (a, n), (b, l), (b, m), (c, l), (c, m), (c, n)} Find the composition of relation (i) R_{1} o R_{2} (ii) R_{1}o R_{1}^{1} Solution: (i) The composition relation R_{1} o R_{2} as shown in fig: R_{1} o R_{2} = {(4, l), (4, n), (4, m), (5, l), (5, m), (5, n), (6, l), (6, m), (6, n)} (ii) The composition relation R_{1}o R_{1}^{1} as shown in fig: R_{1}o R_{1}^{1} = {(4, 4), (5, 5), (5, 6), (6, 4), (6, 5), (4, 6), (6, 6)} Composition of Relations and MatricesThere is another way of finding R◦S. Let M_{R} and M_{S} denote respectively the matrix representations of the relations R and S. Then Example Solution: The matrices of the relation R and S are a shown in fig: (i) To obtain the composition of relation R and S. First multiply M_{R} with M_{S} to obtain the matrix M_{R} x M_{S} as shown in fig: The non zero entries in the matrix M_{R} x M_{S} tells the elements related in RoS. So, Hence the composition R o S of the relation R and S is (ii) First, multiply the matrix M_{R} by itself, as shown in fig Hence the composition R o R of the relation R and S is (iii) Multiply the matrix M_{S} with M_{R} to obtain the matrix M_{S} x M_{R} as shown in fig: The nonzero entries in matrix M_{S} x M_{R} tells the elements related in S o R. Hence the composition S o R of the relation S and R is
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