exam1v2 - EXAM#1 1(5 5 5=15 points Let the following...

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Unformatted text preview: EXAM #1 1. (5+5+5=15 points) Let the following propositions be defined. C: Bill will be rewarded. B: Bill was a hero. M: Mary is telling the truth. Express the following propositions in terms of C, B, and M. (a) If Mary is telling the truth, then Bill was a hero. (b) Bill Clinton will be rewarded only if he was a hero. ((3) Mary is not telling the truth, although Bill was a hero. 2. (5 points) In the Venn Diagram below, shade the region that corresponds to the set (A 7 B ) U (A 7 C). 3. (7.5+7.5+10=25 points) The January 1998 issue of Consumer Reports describes some popular fruit—and—grain bars by the following chart. (Cost is the cost per bar (in cents), Calories is the number of calories per bar, and Fat is the grams of fat per bar.) Product Cost Calories Fat Entenmann’s Multi—Grain Low Fat 34 140 3 Health Valley Fat Free Healthy Tarts 63 150 O Kellog’s Nutri—Grain Low Fat 36 140 3 Health Valley Fat Free Healthy 41 110 0 Betty Crocker Sweet Rewards Fat Free 31 120 O Nabisco Snackwell’s Fat Free 39 120 O Let the domain of discourse U be the products listed in the first column of this chart and let functions from U to the set of natural numbers be defined so that C(13) is the number of calories in a bar of product :1: F(m) is the number of grams of fat in a bar of product m For each of the English language statements given below (i) Rewrite the statement in first order predicate logic using only the functions defined above, and relations and operations of arithmetic. (ii) Determine the truth value of the statement. English language statements: (a) All the products contain some fat. (b) N 0 two of the products contain the same number of calories per bar. (c) If some product contains more fat than all the others, then at least one of the products has fewer than 100 calories. HINT: Use the functions and arithmetic relations to obtain predicates to use in your formulas. For example, F(“Entenmann’s Multi—Grain Low Fat”) 2 3 is true C (“Healthy Valley Fat Free Healthy”) > 120 is false, etc. Give your answer by filling in the chart below. Stat. First Order Predicate Logic Fomula Value 4. (25 points) The symmetric difference of sets A and B, denoted A 69 B, is defined by the equation AEBB={ml (m6AAngB)V(m§ZA/\mEB} If A is a set, show that A ESQ = A. NOTE: To receive full credit, you must justify every step in your argument. Your argument must be well written and your writing must be legible. (This does NOT mean that your argument should be long and verbosell) 5. (30 points) In class I stated the following: “If the function f, with f:A e B, is a bijection, then there is a unique function f‘l, with f‘ch e A, such that f‘1 o f 2 LA and f o f‘1 2 LB.” For this problem you are to show that the converse is also true. That is, you are to show the following: “If the functions f, with ch 4) B, and g, with g: B e A, satisfy 9 o f 2 LA and f o g 2 LB, then f is a bijection. (HINT: Use the equations in the hypothesis to show that f is both injective and surjective.) NOTE: To receive full credit, you must justify every step in your argument. Your argument must be well written and your writing must be legible. (This does NOT mean that your argument should be long and verbose”) Logical Equivalences and Set Identities Equivalence Name Identity p /\ T <:>> p Identity laws A ” U : A p \/ F {I} p A V (D = A p \/ T {I} T Domination laws A e U = U p /\ F 4:) F A o (I) : (I) p \/ p 4:} p Idempotent laws A o A = A pflpfip AoA :A —'—'p 4:} p Double negation law j = A qu<i>qu Commutative laws AUBzBUA pflqfiqflp AflB:BflA (pV9)VT<I>pV(q\/T) (PMWWeWAM/W) Associative laws (AUB)UC=AU(BUC) (AflB)flO:Afl(BflO) pV(q/\?°)<:> (qu)/\(p\/T) Distributive laws Av (800) : (AU B)fl(AU C) p/\(qu)<:>(p/\q)v(p/\r) A”(BUC)=(AflB)U(AflC) fi(p/\q) @fipVfiq De Morgan’s laws A“ B:AUF fi(qu)¢>—'p/\fiq Aszzflg pvape T Excluded middle AUX: U pAfip<i> F Contradiction A"[email protected] ...
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