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Spring 2004 - Popescu's Class - Exam 1

Spring 2004 - Popescu's Class - Exam 1 - \$0 LUﬁON 5 Exam...

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Unformatted text preview: \$0 LUﬁON 5 Exam 1, Mathematics 109 Name: Dr. Cristian D. Popescu Student ID: April 19, 2004 Section Number: Note: There are 3 problems on this exam. You will not receive credit unless you show all your work. No books, calculators, notes or tables are permitted. Good luck ! I. (40 pts.) (1) Use truth tables to show that the following propositional expressions are tautologies (i) (lP—>(Q/\lQ))—>P; (ii) l(R~+S)+~>R/\l3- (2) Use the tautology (1)06) above and other well—known tautologies (eug dc Morgan’s laws) to write the simplest possible form of the negation of the propositional expression (1)“) above. (3) Write the simplest possible form of the contrapositive of the prepositional expression (1X3). _ > _' E _7 4:» 1033“: ('3 4:7 l3 _7 {12 A Rummy—Ev r_ ‘1 agate/Vim ‘, . we \r k 3 UL) WB~>KWEAK1QVC\$3 <—_+_> e» dildo" " L (2:: “all II. (30 pts.) The universe 21 for all the variables in the statements below is the set of integers Z. (1) Prove or disprove the following statement. (3w) 33+(IB+1)33(\$+ 2)3. (2) Prove or disprove the following statement. (V335 9X3 2) \$2 + .112 = 22- (3) Write down the negations of the statements in [1) and (2) above. Ca] we M“ (£9on “M m 1%ﬁ0'k-«H W4}. mm“ @m ) x ”Ha-£1405}! (3+1)? Assam L 3x) kcLM 'xﬁ-thdM) __ (M1) Mir“ ﬁfsrsiszmn “ﬂarBX-nzx-ia III. (30 pts.) (1) Show that for any two subsets A and B of a given universal set U, we have Q) an equality ' A\B=An§, where “3- denotes, as uSual, the absolute complement of B. (2) Assume that the universal set u is the set of natural numbers N. Let A:{p:pprime,p=4k+1, forsomekEN} B = {p : pprime,p=33+2, for some 8 6 N} Use the set—equality in (1) above to show that A\B={p:p=12£+1,forsomeZGN}. A\P_, c Ans (‘9 w» men's/TW’W “e“ A“? Q. A\?.> Qt“) amok Vie-Eh. “A Ky?) Mb“. X%A\B. u... "'— 3 -: '5 .t W =33 E \ \ 1 *E,‘ ' r, k‘mm GM ? S s 1.. “1,171, -. 1m km ea“ _. . uses. ﬁﬂ’k‘th " 31?\W“;:79_L seﬂf‘seﬂ x. ("5” h «QEH 916* ?r M mam—’— I QeH. HM T=RRH1J§W ﬂash M = 3 ? T: 30”)“ ...
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Spring 2004 - Popescu's Class - Exam 1 - \$0 LUﬁON 5 Exam...

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