MATH 211 Final, Summer 2005

en AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH 211 FINAL EXAMINATION SUMMER 2005 Closed Book, 2 HOURS WRITE YOUR ANSWERS ON THE QUESTION SHEET STUDENT NAME | ID NUMBER | Problem 1 2 LIBRARY Ar Preniey MERICAN RY | ’ 1. (10 points) . (a) (2 points} Prove without using truth tables that the implication pgp, is a tautology. (b) (3 points) Prove without using truth tables that the compound proposition prA(PV)-4 is a tautology. (c) (2 points) Let P(x) and Q(x) be predicates on a universal set. Prove the equiva- lence : (vz (P(2) A Q(x) & (Va Plx)) A (V2 Q(z). (d) (3 points) Let P(x) be a predicate on a universal set and R a proposition. Prove the equivalence : (Vx P(x)) V R) = (Va (P(x) Vv R). 2 . (10 points) Consider the function f : A = {—2,-1,0,1} — B= {-1,0,1.2,3}, given by f(a) = a? + Qe. (a) (3 points) Give the range (or image) of A under f. Is f on-to? One-to-one ? (b) (4 points) Consider the subset S of B x A defined by : S = {(b,a)|b= f(a}. Give the elements of (the relation) S. Give your answer in the form of a table (with B horizontal and A vertical). (c) (3 points) Is the relation S a function? Justify your answer.