MATH 211 Exam 1 With Solution, Fall 2004

MATH 211 EXAM ONE Fall 2005 September 29 Name: \C EY The test consists of 2 pages. Justify your work when necessary. (1) Find a simplifying expression for the following sets. Here U is the universe, A and B are two sets such that B C A. a) AUG =A b)AUU = U c)A-U => dAGBA = fH O-A a oh fhANB - BR es)AUB =f h)ANB = & ASB = ALB PO) . 5 $4 (2) Using only p, g, r, > and/or the connective A, write a proposition equivalent to each of the following Qaraar £4 ype) — ¥ VPaATtd) +r “1(AC pana) ve = (Palaver = a(gntg) ary) (b) p> lg r) onion Be i“ aN (49) =1Pv(14vr)21(p A\4v r)) =P l(t a) (3) Write the contrapositive and converse of the statement: “You sleep late if it is Saturday”. the statement itsel§ LS & if it is Sat «en yeu slee Sp centra. us * iS yeu dewt sleep we Hon it is not Saf.” Converse is > “ af You sleep wet Han akis Sat.” (4) In the following, P(xTy) means “a2 + 2y = wy”. Where x and y are integers. Determine the truth value of the statement. (a) T 3 SyP(z,3) this is not even a proposition, y is free. (b) T VelyP(z,y). whew X22 We get Ce we YN (c) TF) S2vyP(2,y) whn xX=0% 2ZH=0 Dow Cam o Wg Scum ww « Sar J (5) Suppose the variable « répresents students~and tl id ane ° which is Aye WY carton ly, mM choice ¢ ee ne varieble y represents courses, and A(y): y is an advanced course S({a): © is a sophomore F(x): «isa freshman T(#,y): # is taking y. Write the following statements using these predicates and any needed quantifiers. (a) There is a course that every freshman is taking. ByVe( Fcx) T(x) (b) No freshman is a sophomore. vx( ech ee ii | S@)) ar 78x ( Fc Xm) A Sx) (c) Some freshman is taking an advanced course. Bx (F(x) AT(x,y) 0 AY?) (d) There are at least two freshman students taking the exact same courses. a x, a ( FC) A F(%2) Aa Vy (7 (wy) <= Tx2y))) (6) Determine whether the following argument is valid. = nok valid ; vie assignment =(p V q) = PeTy SEP & PST (7) Determine whether the following argument is valid. 1) She is a Math Major or a Computer Science Major. 2)If she does not know discrete math, she is not a Math Major. 3) If she knows discrete math, she is smart. 4) She is not a Computer Science Major. 5) Therefore, she is smart. G): (1) &CG) medn she iy @ MATH mejor ( disjunction syllogism) 7) TS she isa math mapol Fun she Knows discrete Coon brapesi hve of 2), x) ©) % (CF) &% modus PenensS meow she Knows discrete mah. 9) (3418) & modus poriw give G) se it ts) O valid argument (8) Determine whether the rule describes a function. If your answer is no say why. (a) f: N- N where f(n) = Yn. ne beCawe VN weed not be im INL e.g. ne2. (b) 9: NN where g(n) = any integer > n. Ne, Phe defi ni hon Ls net mint) ecg, 4 (2) Ceuta Le 33 bg Dy eres (9) Give an example of a function from @ to N that is both one-to-one and onto. oat Z —»> IN seo ee . ( —+ | $Cz) = 3 ae[: Gags Be 2z-) if Z>o a -Z2—» 4 -2zZ if Z2<o = © [oe © (10) Give an example of a function from Z to N that is onto but NOT one-to-one. $ (2) = \z| Has takes wey twee opposite Valuer jt Me Vole , hun @ not ons — to one Lut onte ( eve osikve rnhue iy He eae ee of he Y (11) Let f: A+ B. Let B’ Cc B. Show that f(f-'(B")) CB". WHAT condition is needed for the containment in the other direction? tk 4 ¢ Cree) > xe £8) such thet 4 = sae) ‘ Qu since x € 508") esas ack Vad ¥ (x) e®, Gathuring We foot twe piece Q, c vena) We need ay eloanent &% 8 te be an one of Saw ened of ar This ts quacanteed whom \ es ion ;