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Unformatted text preview: f} NAME: 2 ’3 ' Prof. Girardi 04.27.06 Final Exam Math 300 —_ MARK BOX PROBLEM POINTS Problem Inspiration
1 I— 11 _l— Ease handout
2 12 class handout
3 5 I § 1.2 # 7e
4  1st chokgl 9 4a: hand—in HW § 1.5 # 3g 41): lookat HW § 1.6 # 7k
4c: iii—class HW 1.7 # 3d
4d: Theorem 2.6 n 4  2nd choice 9 —l
_4  3rd choicd— 9  
F TOTAL 55 _l_  INSTRUCTIONS: (1) The MARK BOX indicates the problems along with their points.
Check that your copy of the exam has all of the problems. (2) When applicable put your answer on/ in the line/box provided.
Show your work UNDER the provided line/hex. If no such line/box is provided, then box your answer. (3) For problems that request a proof, write a neat FORMAL proof and use only logic and the
definitions of the concepts involved (i.e., do not quote a problem from the book). You may
include your skeleton, if you so wish. (4) During this exam, do not leave your seat. Ifyou have a question, raise your hand. When
you finish: turn your exam ever, put. your pencil down, and raise your hand. (5) You may not use a calculator, books, personal notes. (6) This exam covers (from A Mnsition to Advanced Mathematics by Smith, Eggen, and St.
Andre: nth ed.): Sections 1.1., 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 2.1, 2.2 . Numbers: a real numbers = R 0 positive real numbers 7— ll?“ : {:12 E R: :1: > 0} 0 natural numbers = N = {1, 2, 3, 4, . . o integers = Z = {0, ::1, 2, ::3, 4, . . .} o rational numbers = Q = {E E "t: p, q E Z and q 7é U} Throughout this exam: (1) P, Q, and R be propositions. (2) P(:r:1, . . . , 11:7,), (201:1, . . . , 311,), and R(:rl, . . . ,mn) are open sentences with variables 51:1, . . . ,atn. 1. Make true statements by ﬁlling in the blanks with the appropriate number between 1 and 11 from below. Use each number once, and only once. m (P => mm? => 1:)
pm? = Q)/\(Q =¢ p)
,6) ~ Q => ~ P MUD/x m Q) => R M (M ~ Q) in ( ~ P) A (~ 62) {WP => Home =2 R)
£8) P =~Q l0? (Vx)[~ Pm] ()0) (39:)[~ FLT)1 (11) (EmJlPWJ/WWNPW) => :6:le P => Q is equivalent to 3
P <=> Q is equivalent to 2
m (P = Q) is equivalent to w (P A Q) is equivalent to m ( P V Q) is equivalent to G) ____
P 2 (Q A R) is equivalent to P => {Q V R) is equivalent to (P V Q) i R is equivalent to "
m [(Vm)P(m)] is equivalent to I}
~ [(3:r)P(.r)] is equivalent to 9
(3!:c)P(:n) is equivalent to l l 2' Circle T if and only if the statement is TRUE. Circle F if and only if the statement if FALSE. 1. @ F (Fm)(vy)F(x,y) :> (chvmumy)
2» F (FFMFMM) => (ayuamxw) F {Var)P(:F) (Elm)P(m)
’4 T if? (3F)P(rc) => (mph)
<3 T (VF)[P(F)VQ(F)] =» [(ijmmwwmxm
3 F [(vmwmwwxmtxn => wasnptxwmxn
L1 F (WM) :> em] => mm.) => (VFWCFJI
“1 T [(vx)F(ar) => (qume = MW) => mam F F‘ (Mammy => (EFH‘F’mey)
{:1 F (ax)(v:u)F(x,y) => (Fyxaxjpw F (Vm)[P($)/\Q(m)] => [(VF)P(F)MVF)Q(F)]
'3 F [wanmeme => (VF)[P(F)/\Q(F)] 3. Show that
N (P = Q) is equivalent to (PA N Q) by making a truth table and then writting a sentence or so about how you are remng your truth table to come up with your answer. 4. Do THREE problems from problems: 4a, 4b, 4c, and 4d. Fillin the blanks on this sheet of paper but include your proof parts on a separate sheet of paper (with your name on each sheet please). Circle THREE: I am doing problems 4a , 4b , 4e , 4d 4a. Let :1: E Z. If 8 does not divide x2 — 1, then m is eveﬂ
a. Let a, b, e E Z. By deﬁnition: . 5‘: l' v.' I . '\ a divides b, denoted by elb, ifand only if [ 3 ' A l l i ‘ " J
J ‘ “I: j I) n f l, w; .l
c is even if and only if l 3 r‘C 51.! j l a. ’ “' “‘ x cisoddifandonlyif (Cr 9? o. Symbolieally write, using quantiﬁers, statement 4a. ., ’ ‘Ir’l f I: _
' ‘ i’ , '1 \' a}r >5 .r' *. ' .u l w 4“ m .
1} E c “I Ii =_\ .2 ' _ ..— , ,. Ml seesaw o._ Prove statement 43.. o. By deﬁnition, an integer M is odd if and only if '  ' : 31'“? i 0. Symbolically write, using quantiﬁers, statement 41). I I
:' E M E“ l ‘r :7 . :' fr" 7"; 5%.: (f 0" O I} \ o. Prove statement 4b. 0: 40. For each rational number :1: and each irrational number 3;, there is a unique irrational
number 2 such that y + z = r. 0. Let a and b be real numbers. By deﬁnition: ” . 4 ...' I ._ Ir
 a is rational if and only if ( 3 l3 Ii 9: .' l ‘ b is irrational if and only if o. Synibolieally write, using quantiﬁers, statement 4e. .1“! ' l a" i ‘11 v. .2 ' _ l 31. '7: a lit» 11L) .. if 3""— ' _—L L_. o. Prove statement 4e. You must carefully prove, using only the deﬁnitions above. (and not using another homework problem), that your 2 is indeed irrational. 4d. Let A, B, C be subsets of the universe U. Show that A U (B ﬂ 0) = (A U B) n (A U C) o. By deﬁnition: :‘i'i :1: e A o B if and only if .;r rt x mEBﬂCifandonlyif a. Let P, Q, and R be propositions. The disbributive rule says that (Mm) v ("H R: [P A (Q V R)] is logically equivalent to [P v (Q /\ R)] is logically equivalent to (I? V Q) A ( v R) o. Prove statement 4d. You can use the logical equivalents for propositions that you just wrote above. I“. Ff 1.? W wwﬂ—
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