Recall the relation composition operator ; defined as: R, R = {(a, c) : there is...

 

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Recall the relation composition operator ; defined as: R, R = {(a, c) : there is a b with (a,b) R and (b, c) = R} For any set S, and any binary relations R, R2, R3 C S x S, prove or give a counterexample to disprove the following: (a) (R; R); R3 = R; (R2; R3) (b) I; R = R; I = R where I = (c) (R; R) = R; R (d) (R UR); R3 = (R; R3) U (R2; R3) (e) R; (R R3) = (R; R) (R; R3) {(x,x) : x S} (4 marks) (4 marks) (4 marks) (4 marks) (4 marks) Recall the relation composition operator ; defined as: R, R = {(a, c) : there is a b with (a,b) R and (b, c) = R} For any set S, and any binary relations R, R2, R3 C S x S, prove or give a counterexample to disprove the following: (a) (R; R); R3 = R; (R2; R3) (b) I; R = R; I = R where I = (c) (R; R) = R; R (d) (R UR); R3 = (R; R3) U (R2; R3) (e) R; (R R3) = (R; R) (R; R3) {(x,x) : x S} (4 marks) (4 marks) (4 marks) (4 marks) (4 marks)