4) In this HW, set, N = { 1, 2, 3, ) 5) al rES~y whereis an equivalence relation Exercise 1. Let z e R \ Z. Prove that(-z-- -1. What happens if we consider e R? Exercise 2, let R be the relation defined by :?Ry iff 2r + 3y is a multiple of 5, Here z, y Z a) Is rRr for any integer a b) When rRy, do we have yRr, Va, y Z? c) Prove that when Ry and yRz then R,r, y, z EZ d) Is R an equivalence relation? Exercise 3. let R be the binary relation defined by : zRy iff r, ?? is divisible by 3 where z.y E Z. IR an equival relation? Exercise 4. Let (z, y) E R. R., and define?~ y to mean-E Q a) Prove that is an equivalence relation b) Show that v3-12 c) Show that [vgjnl Exercise 5. Define z ? y to mean Ir|-11 when z, y R a) Verify that is an equivalence relation, and confirm that tol = {0} b) Let S be the sets of all equivalence classes of . -, and define ? : S S defined? i.e. is A a function?) + S by letting ?([a],PD-lo +&J 18 ? Exercise 6. Define f : A ? ? by f(z) = z2 + 14x-51. Determine whether f is one-to-one and whether it is onto in c of the following cases: a) A N and B-(be Z: b -100) b) A- R and B bER:62-100 Exercise 7. Determie whether each of the following is a one-to-one and/or an onto. Give a proof or provide a coun example to justify your answer =?for all integers n. Exercise 8. Prove that Exercise 9. Dehise f: Z+ N bs )20 a) Show that f has an inverse. b Find(2586) Exercise 10. Let A R, and suppose f : A ? A is a function with the property that/-1(r)-(f(z))-1 for all z E A