Question

specifically on finite

i pmu r the number of objøcts or ways. Leave your answers in fornsiala form, such as C(3, 2) nporkan?(2) Are repeats poasib Two points each imal digits will have at least one xpeated digin? I. This is the oounting problem Al ancmher so ask yourelr (1) ls onder ipo n How many strings of four bexadeci ) A Compuir Science indtructor has a stack of blue can this i For parts c, d. and e, suppose that in a bushcl of 100 apples there are 20 thnt have worms (c) How many ways can a group of 13 apples be selected tor a pie? maper, a sack of gellow paper, a stack of red paper, and a stack of green paper. la how many ways instructor select a group of 25 sheets from these piles to prin the fromt covers of the final lnbow many ways will all İSapplesthe pop selected have n how many ways will at least one apple (of the 15) have a worm I be . trwh table todetermine the logoal equivalence of(pvg) A (p-小(g-r-r l' this . ind the flaw in each proof Clearly describe the flaw. (a and &-3 points each, 4 points) e) Prove 1 for all onnegative integers n, whenever a is a nonadro real number Basis nep1 is true by algebra and the definition that a 0 ldeteve sp Assumed» 1 for all nonneganve integers k with 1. Then ak+1 mclusion: a1 for all nonnegative integers n, whenever o is a bonzero real namber -T- 4 Consider the conditional statement "Wit a Friday, then it m (a) O points) Write (in words) the contrapositive of this statement (c) (2 points) Write (in words) the inverse of this statement S. Use mathematical induction to prove 6 Prove If ab and altb + e) then ale (3 points) Write (in words) the negation of this statement 2 poins) Write (in words) the converse of this statement that for n 2 1, 1-11+2-21+3-3++n-n(n+1)-1 7.Pe that for all sets A and B, PIAnB) P(A)nPB) (a) Find the Boolean expression that coresponds to the circuit on the right (b) Construct the circuit foe the Boolcan function AND 9. (a) Represent 9s in binary (base 2). (b) Represent-98 in 8-bit teo's OR 12. A solution to an recursive 13. This is a counting problem. All questions in this peoblem ask you for the pumber of objects or ways. Leave your answers in formala form, such as (G). Ask younscdf Is order important? Are repeats posnblie? relation is given by the equation + where ag2 and a-7. Find the exsplicit formula for a 14. (a) A finide-state automation is given by the annotated next-state table (b) sketch a state diagram (c) Contruct the circuits that impliment this automata IS. A copy machine techincain get a bonus if he or she services 50 or more copy machines in one week. Suppose due to a computer failure you can oely find the you serviced 198 copy machines daring a four weck period. How can you prove that you deserve the bonus for one week? Can you prove thet you deserve the bonus for two weeks? AuC 21. Make up your own sets then find (a) Are A _ B from part c .nd C di B, as follows: AeB-(A-Bu(B-A) and B, denoted A 22 (10 points) Given any sets A and B, define the symmetric difference of A Prove that A B-(AUB)-(An B) 23. Suppose that you run out of time on a multiple choice test and answer the lest two and- r Lat 'ad' indicate that you choose for the next to last questionanddfor the last quaka (a) List the elements in the sample space whose outcomes are all the posible answers (b) If cach solution is equaly likely to occur, then for each question there is a chance (c) What is the probability thas at least one solution is correct 24 Radio station identification in the United States consist of 3 or 4 cal-lenerse wre the first Icticeris either a K'or W and the other letern can be any other letter. How many different call-leters are possible? 25. Six people atiend the theater together (a) How many ways can they be seated in a row? paged. How many ways can the people be seated in a row if exactly one of the seats is on the aisle and the doctor is in the aisle sear? 26 A program development 1cam has 14 members, 5 women and 9 men (a) How many ways can a group of seven be chosen to work on a (b) How many teams of seven can be chosen that don't contain any women? (c) How many leam of seven can be chosen that contain 3 women and 4men? questions randomly. If each question has four possible answers,Y, v.e that your answer is correct and a-chance that both are conect (b) Suppose one of the six is a docior who must sit on the aisle in case she is 27. How many integer solutions are there to the equation zi + z+ + 10 (a) if each z 2 1? (b) if cach z22c) if each23 28 (a) How many functions ะethere tam a set with three elements to a set with four elements? (b) How many one-to-one functioes are there from a set with three clements to a set with four clements? 29 Design a finite-state automation with input alphabet equal to (0, 1) that acpepts the set of all strings that start with 01 or 10. (a) Is F one-to-one? Is it onto? Can you disprove or prove? b) is G one-to-one? Is it onto? Can you disprove or prove? Input Current 14. (a) A finite-state automation is given by the annotated next-state table. → |801180181 (b) sketch a state diagram. (c) Contruct the circuits that impliment this automata. State 83 1 83

Answer 1

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