6 YORK UNIVERSITY SC/MATH 1019 3.0 D WRITTEN ASSIGNMENT #2 6. (10 points) Let R be a binary relation on a set A that is reexive and transitive. Prove that R\" = R for all positive integers n. 7. (5+5 points] (a) Show that a binary relation R on a set A is antisymmetric if and only if R 0 R'1 Q A where A = {{o,a)|a E A} is the diagonal (equality) relation . (b) Let R be the binary relation on a set A = {(11, a2, {:3} represented by the matrix 0 1 0 NIB: 0 1 1 1 0 1 Find the matrix DAR1 representing the relation R'l. YORK UNIVERSITY SC/MATH 1019 3.0 D WRITTEN ASSIGNMENT #2 8. (5+5 points) (a) Suppose A is a nonempty set and f : A - B is a function on A. Let R be the binary relation on A defined by R := {(x, y) E A X A f(x) = f(y)}. Show that R is an equivalence relation on A and describe the equivalence classes of R. (b) Give a description of congruence classes modulo 5 on Z (i.e. the equivalence classes of the congruence equivalence relation Rs := {(m, n) EZ x Z |m = n (mod 5)})