Problem 11. Represent cach of these relations on (1.2,3,4) with a matrix and a digraph. (b) R2={(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)) Problemi 12. Let R be the relation on Z#x Z+ such that ((a, b), (c,d)) E R if and only if a + d-b+c. Show that R is an equivalence relation. Problem 13. Let X -(1,2,3,4,5), Y (3,4 and C - (1,3). Define R on P(X) by (A, B) ER if AUY- BUY (a) Prove that R is an equivalence relation. (b) List the elements of IC), the equivalence class containing C. Problem 14. Let X (0,1,2,...,14). Define a relation R on X by (oD) ER it 512-v for any a,VEx. (a) Show that R is an equivalence relation. (b) Determine the distinct equivalence classes for this relation R on x. (c) Construct a digraph for the relation R Problem 15. Define a relation R on RRIRR is a function) by (f.9) ER if f(0)-9(0). (a) (b) Prove that R is an equivalence relation. Let f(x) a for all e R. Describe [f). lations on a set X, show that R1 n R2 1s an equivalence