Question

Let a function where f:Z5 → Z5 defined by f(x) = x3 (mod5). a. Is f an injection? Prove or provide a counter example. b. Is fa surjection? Prove or provide a counter example. c. Find the inverse relation of f. Verify that it is the inverse, as we have done in class. d. Is the inverse of f a function? Explain why it is or is not a function.

Formal proof and state which proof style you use

Answer 1

fexs = x3 cmeds) ab) Here, f is defined by, .: fcl) = 3 (mod 5) = 1 cmods) f(2)= 23 cmods) = 8 cmods) = 3 cmod s) f(3) = (mod 5) = = 27 mods) Ē 2 cmod 8) FC24) = 43 (mod 5) = 64cmods) 4 cmod 5) 03 (mods) o cmods) 33 fco) = aby Here fon = 1 cmod s), f(2)= 3cmod s) f(3) = remods) uomod so fco) = ocmod s) f(4) = Here , for different values or f(x) we have unique value of x. ie. for f(x) = foxz) x = x2 f is one-one (codomain) by Here, Every element of 5 is image of some element in Zs .C Domain) ie. All element, {0, 1,2,3,4} in 25 are image (corondomains or some element in as (Romain). - Range f = Condomain off f is onto-

g How, since, fcl) = 1 cmods) = f'(o=1 cmods) f(2)= 3cmods) =) F1 (3) remods) f (3) = 2 cmol 5) = f(2)= 3 mods) f(h) = hemods) = f'(a)= 4 mods) f(c) = 0 cmods) f-100-ocmods) Mow, (for) = folf (0) = f(1) = lamod 3) (for) (2) = folf (2)) = f(3) = 2 anods) (For (3) = fo (f-1(3)) f(2)=3 Chods) (for) 4) = folf-(4) fo (4) = 4 cmod s) (foplo for fico) - fcoj - ocmods) similarly, (flofcl) = 1, (flot (2) = 2 (f tof) (3) = 3 , (f-tof (4) = 4 (ftof) (8) = 5 (ft of) = fot = I ft is inverse or is f is verified. di kes it is function, (Because it delined fer every relue in ag and for different values of x f(x) is different which we can see above . alf is a function.