13. a) Consider f; Z+ → Z+ define by f(a) =a2. Check if f is one-to-one and / or into using suitable explanation. b) What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. Is R a partial order. c) Define a lattice. Explain its properties. 14. Show that if eight people are in a room, at least two of them have birthdays that occur on the same day of the week. 15. a) Give a relation which is both a partially ordered relation and an equivalence relation on a set. b) Let P be the power set of {a, b, c}. Draw the diagram of the partial order induced on P by the lattice (P,,). 16. a) Let A,B,C єR2 where A = { (x,y) / y = 2x + 1} , B = { (x,y) / y = 3x} and C = { (x ,y) / x - y = 7} . Determine each of the following: i. A B ii. iii. iv.