MATH 211 Exam 2, Fall 2003

Math 211 Quiz I ( Fall 2004 ) Time 75 min Name: ID#: Circle your problem solving section number below : Section | 12:00 F Section2 1:00 F Problem 1 1-LOGIC Define the connective (| ) by p | g_ is true when both p and q are false and it is false otherwise. Draw the truth table for this logical operation then prove or disprove the logical equivalence of p | (gar) and (pyq)vr (8 pts) 1. D a iy 7 D y y) 1 wore pd 1 agQané f A _ q » 19 2 0 } | | “| 9 { 9) | | 0 nt { ? 0 | i 0 3 | ! { 9) 0 0 (40 C 8 0 ; (6) 1} do |} dp y 9 6 a ui |e Oo , , 10 (ft 4 ! | | ) i i (5 pts) 2. Determine whether the following compound proposition is a tautology. [bv -g)s@or)Cg>rer (4 pts) 3, Translate: a. The English statement "The product of two positive numbers is positive’ into a logical expression . then write the corresponding negation. t b. Into English the logical expression: Wrdr(x +p2e2a2x-yea 1) Lt [ da-gel I-PROOFS (6 pts) 4. Show that 5 is an irrational. Specify what method of proof you are using? (S pts) 5. Prove that the sum of a rational number and an irrational number is irrational. Name your method of proof. I1I-FUNCTIONS (7 pts} 6. The function f:/N x /JN——+IN is defined by f(x, 1°) = 2°6". a. Is fone to one? Justify your answer Alito de \ Reco = £0. 4) “4 xy Ye ; - — ROK Y= 2 © _. ) ey —~e yn a ecw inte) 2 uet he Yeo} Ky \ Vv & = b. Is f onto? Justify your answer ec Is / invertible? Justify your answer (7 pts) 7. Prove that: | + A =n for all integers 7. 1V-RELATIONS AND EQUIVALENCE RELATIONS (9 pts) 8, Form, in IN define m= 7 if m? ~n7 isa multiple of 3 a. Show that = is an equivalence relation on IN. b. Describe (0] and [1]. ce. Show that [0] and [1] are the only equivalence classes for ~? Justify your answer. V-POSETS (9 pts) 9. Consider the sct S={a,b,c}. Let P(S) be the power sct of S. a. Describe the elements of P(S). b. Let C be the inclusion relation on P(S). Show that (P(S),G) is a poset. ce. Draw the Hasse diagram of this poset. d. Give the minimal and maximal elements of this poset.