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# Discrete Structures CSI 2350

Baylor University

GPA 3.85

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This 18 page Class Notes was uploaded by Melvin Bednar on Saturday October 3, 2015. The Class Notes belongs to CSI 2350 at Baylor University taught by Peter Maurer in Fall. Since its upload, it has received 62 views. For similar materials see /class/217936/csi-2350-baylor-university in ComputerScienence at Baylor University.

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Date Created: 10/03/15

1 Graphs a Terminology De ne Vertex Node Edge Arc Path Adjacent Simple Path Cycle Simple Cycle b Relations Show the graph of the following relationship ab bc bd be cf cg eh c Euler Paths Draw a graph that has an Euler cycle Draw a graph that has an Euler path Draw a graph that has neither d Bipartite What is a bipartite graph 2 Counting a Multiplication principles I have three urns with numbered balls in them The rst has three red balls the second has four green balls and the third has two white balls How many ways are there to choose one white ball one red ball and one green ball b Unions and Intersections LA Permutations and Combinations a Binomial theorem in x l20 what is the coef cient of x b Algorithms What is the next permutation after 54123 Five things taken three at a time what is the next combination after 135 Probability a Equally likely events Suppose I roll two dice and ip two coins What are the equally likely outcomes of this experiment b Adding disjoint sets If we choose a number n from 1 through 100 what is the probability that n E 0mod5 What is the probability that n E 2 or 3mod 5 c Adding nondisjoint sets If we choose a number n from 1 through 100 what is the probability that n is divisible by 5 What is the probability that n is divisible by 2 What is the probability that n is divisible by 5 or by 2 4 5 Prove that Z4i7 2n2 9n 11 6 Prove that 5n1 1 51 4 gt1 9 0 Multiply the following 1 1 1 2 1 2 0 2 0 1 1 2 Multiply the following 0 2 3 1 0 1 3 1 0 1 1 0 1 0 1 0 1 2 Combine the following 123456123456 456123321654 How fast are the following algorithms BubbleSort Quicksort Mergesort Repeated Minimum sort Insertion sort Convert from binary into hexadecimal 11101011101001 1001011101110 101010011 1000000100001 Give addition and multiplication tables for Modulo 5 arithmetic Which of the following are propositions Eat at Joe s Make me smile I m in love with a big blue frog If cows were green then you couldn t see them in the grass Did you eat yet Money is the root of all eVil Wop90 Which of the following are true a If cows are green then pigs can y If Xgt7 then X5gt12 If 3 is a prime number then 10 is odd If I won the lottery then I will go to Paris If9 is a prime number then 10 is bigger than 100 0305 V39 0010 0 Find the minimal Boolean Equations for the Given U12345678910 A123 B456 C678 give the following a A n B b A U B c B n C d A e B C State and prove De Morgan s laws for sets Give the commutative associative and distributive laws for Boolean Algebra Give the identity and complement laws for Boolean Algebra What is an equivalence relation What is a partial order Negate a Vxx gt 7 b Vxx lt x 1 c EIxxlt3xgt7 Prove that x13 x3 3x2 3x1 truth tables 1010 24 Convert the following expression to pre x notation abcefcdeab CSI 2350 N E 4 V39 0 gt1 9 gt0 Review Nov 1 2004 Prove that Z2iln22n 11 Prove that 3n1 1 E3 2 Prove that n3 gt 2n2 l whenever n gt 2 Multiply the following 2 1 1 2 1 3 1 2 Multiply the following 2 l l 2 1213 Multiply the following 2 1N 1 3 1 2 l 12 2 at Multiply the following 1 2 3 3 3 1 L3 2 1 L2 1 1 l 2 l 3 l 2 How many permutations are there on a set of 3 elements 4 5 6 Combine the following permutations as indicated l 2 3 4X l 2 3 4X 3 l 2 4 f2 1 4 3 CSI 2350 10 N LA UI ON 00 O N O N N N N 4 Review Nov 1 2004 Combine the following 1 2 3 4X 1 2 3 L4 3 2 1 f4 1 2 3 Combine the following 1234123 L31421 tz134 Combine the following 1234123 L4132J t1234j What is a function What is an onto function What is a onetoone function What does it mean to say that a function is recursive How fast are the following algorithms BubbleSort Quicksort Mergesort Convert from binary to decimal 1011011 1010 11101110111010101011101110 Convert from decimal to binary 43 6006 2040 1010 2048 Convert from Hexadecimal to binary 1A3F FF 401 AA55 1011 Convert from binary into hexadecimal 11101011101001 1001011101110 101010011 1000000100001 Give addition and multiplication tables for Modulo 3 arithmetic Give addition and multiplication tables for Modulo 4 arithmetic Give addition and multiplication tables for Modulo 5 arithmetic Does the inverse law hold for Modulo 6 arithmetic Modulo 7 Modulo 9 1 Graphs a Terminology De ne Vertex Node Edge Arc Path Adjacent Simple Path Cycle Simple Cycle b Relations Show the graph of the following relationship ab bc bd be cf cg eh c Euler Paths Draw a graph that has an Euler cycle Draw a graph that has an Euler path Draw a graph that has neither Advanced What is an articulation point What is a biconnected graph What does it mean to say that two graphs are isomorphic What is a planar graph What is the graph coloring problem Know how to nd a shortest path 2 Counting a Multiplication principles I have three urns with numbered balls in them The rst has three red balls the second has four green balls and the third has two white balls How many ways are there to choose one white ball one red ball and one green ball b Unions and Intersections 3 1 3 Permutations and Combinations a Binomial theorem in x l20 what is the coef cient of x18 b Algorithms What is the next permutation after 54123 Five things taken three at a time what is the next combination after 135 4 Probability a Equally likely events Suppose I roll two dice and ip two coins What are the equally likely outcomes of this experiment b Adding disjoint sets If we choose a number n from 1 through 100 what is the probability that n E 0mod5 What is the probability that n E 2 or 3mod 5 c Adding nondisjoint sets If we choose a number n from 1 through 100 what is the probability that n is divisible by 5 What is the probability that n is divisible by 2 What is the probability that n is divisible by 5 or by 2 5 Prove that Z4z 7 2112 9n 11 ON Prove that 5n1 1 51 4 gt1 Multiply the following 1 1 1 2 1 2 0 2 0 1 1 2 Multiply the following 0 2 3 1 0 1 3 1 0 1 1 0 1 0 1 0 1 2 9 50 Combine the following 1 2 3 4 5 6 1 2 3 4 5 6 4 5 6 1 2 3 3 2 1 6 5 4 10 How fast are the following algorithms BubbleSort Quicksort Mergesort Repeated Minimum sort Insertion sort 11 Convert from binary into hexadecimal 11101011101001 1001011101110 101010011 1000000100001 12 Give addition and multiplication tables for Modulo 5 arithmetic LA Which of the following are propositions Eat at Joe s Make me smile I m in love with a big blue frog If cows were green then you couldn t see them in the grass Did you eat yet Money is the root of all evil rm 999 14 Which of the following are true a If cows are green then pigs can y b If Xgt7 then x5gt12 22 23 ABCD F 0000 0 Given Ul2345678910 c If 3 is a prime number then 10 is odd d If I won the lottery then I will go to Paris e If 9 is a prime number then 10 is bigger than 100 Al23 B456 C678 give the following A n B A U B B n C A B C rye96x State and prove De Morgan s laws for sets Give the commutative associative and distributive laws for Boolean Algebra Give the identity and complement laws for Boolean Algebra What is an equivalence relation What is a partial order Ne gate a Vxx gt 7 b Vxx lt x l c EIxxlt3xgt7 Prove that x l3 x3 3x2 3xl Find the minimal Boolean Equations for the following truth tables ABCD F ABC G 1000 0 000 001 010 01 l 100 101 l 10 l l l oool OD ID ID I CSI 2350 N 4 gt1 WWFDPQPQFF Review Sept 20 2004 Which of the following are propositions My feet are too big Your feet are too big What time is it It s time for dinner Well then bring me a steak Mimsy were the borgroves I like movies Of time and a river Some say the world will end in re Give truth tables for the following a p v q b p v q r 0 PA9VPV Which of the following are true a If cows are green then pigs can y b If Xgt7 then X5gt12 c If 3 is a prime number then 10 is odd d If I won the lottery then I will go to Paris e If 9 is a prime number then 10 is bigger than 100 Use truth tables to determine which are true and which are false 21 p gtqAq gtVEp gtqAp gtr b pAq gtVEp gtqAp gtr 0 p gtqAp gtVEp gtqr Is the following inference rule correct or incorrect PAq F pqur Given Ul2345678910 Al23 B456 C678 give the following a AnB b AUB c BnC d A e B C State and prove De Morgan s laws for sets Give the commutative associative and distributive laws for Boolean Algebra CSI 2350 gt0 Review Sept 20 2004 Give the identity and complement laws for Boolean Algebra What is an equivalence relation What is a partial order What is a lnction Ne gate a Vxx gt 7 b Vxx lt x l c EIxxlt3xgt7 Give the commutative associative and distributive laws for integers Give the identity and inverse and cancellation laws for integers Prove that x2 2x l x l2 CSI 2350 N L 1 UI ON 1 Review Mar 29 2005 Prove that Z4i 72n2 5n 11 Prove that 3 3 2 5 Z z 1 quot quot quot 11 3 Prove that n3 gt 2112 1 whenever n gt 2 Multiply the following 2 12131 2 Multiply the following 2 1 11213 2 Multiply the following 1 3 M P 12 2113 j 2 Multiply the following 1 2 3V3 3 1 L321JL211J 121312 Combine the following permutations as indicated 12341234 3124 t2143 CSI 2350 Review Mar 29 2005 0 N N Combine the following 1 2 3 4X 1 2 3 4 L4 3 2 1 f4 1 2 3 Combine the following 1 2 3 4X 1 2 3 4 L3 1 4 2 f2 1 3 4J Combine the following 12341234 L4132J t1234J Convert from decimal to binary 2 s complement notation All results will be 8 bits long 10 12 37 14 100 97 Convert from decimal to binary signed magnitude notation All results will be 8 bits long 10 12 37 14 100 97 Convert from Hexadecimal to binary 1A3F FF 401 AASS 1011 Convert from binary into hexadecimal 11101011101001 1001011101110 101010011 1000000100001 Give addition and multiplication tables for Modulo 3 arithmetic Give addition and multiplication tables for Modulo 4 arithmetic Give addition and multiplication tables for Modulo 5 arithmetic Does the inverse law hold for Modulo 6 arithmetic Modulo 7 Modulo 9 Convert the following to pre x notation a b c d a b c d a b c d Convert the following to suffix notation a b c d a b c d a b c d CSI 2350 N 4 gt1 WWFDPQPQFF Re ew Feb82005 Which of the following are propositions My feet are too big Your feet are too big What time is it It s time for dinner Well then bring me a steak Mimsy were the borgroves I like movies Of time and a river Some say the world will end in fire Give truth tables for the following a p v q b p v q r 0 PA9VPV Which of the following are true a If cows are green then pigs can y b If Xgt7 then X5gt12 c If 3 is a prime number then 10 is odd d If I won the lottery then I will go to Paris e If 9 is a prime number then 10 is bigger than 100 Use truth tables to determine which are true and which are false 21 p gtqAq gtVEp gtqAp gtr b pAq gtVEp gtqAp gtr 0 p gtqAp gtVEp gtqr Is the following inference rule correct or incorrect PAq F pqur Given Ul2345678910 Al23 B456 C678 give the following a AnB b AUB c BnC d A e B C State and prove De Morgan s laws for sets Give the commutative associative and distributive laws for Boolean Algebra CSI 2350 Review Feb 8 2005 9 Give the identity and complement laws for Boolean Algebra 10 Prove that A U g A EB Ql 6f 11 Prove using Boolean algebra abcaccc CSI 2350 N E 4 V39 0 1 9 Re ew Sept192005 Convert the following numbers into 8bit binary numbers using 2 s complement notation 10 100 127 32 80 64 1 128 Convert the following binary numbers into decimal These are 8bit binary numbers in twos complement notation 10101010 0101010111011011 00001111 00110110 00010011 00000011 Convert the following hexadecimal numbers into binary 3A4 2A89 B 341 22 0FE 01D Which of the following are propositions My feet are too big Your feet are too big What time is it It s time for dinner Well then bring me a steak Mimsy were the borgroves I like movies Of time and a river Some say the world will end in fire wrung9999 s Give truth tables for the following a p v q b p v q r 0 PA9VPV Which of the following are true a If cows are green then pigs can y If Xgt7 then X5gt12 If 3 is a prime number then 10 is odd If I won the lottery then I will go to Paris If 9 is a prime number then 10 is bigger than 100 DQOU Use truth tables to determine which are true and which are false 21 p gtqAq gtVEp gtqAp gtr b pAq gtVEp gtqAp gtr 0 p gtqAp gtVEp gtqr Is the following inference rule correct or incorrect r quVV CSI 2350 Review Sept 19 2005 50 Given U l2345678910 Al23 B456 C678 give the following A n B A U B B n C A B C rue99 10 State and prove De Morgan s laws for sets 11 Give the commutative associative and distributive laws for Boolean Algebra CSI 2350 N E 4 V39 0 gt1 9 gt0 Review Dec 6 2004 Prove that Z4i7 2n2 9n 11 Prove that 5n1 1 51 T Multiply the following 1 1 1 2 1 2 0 2 0 1 1 2 Multiply the following 0 2 3 1 0 1 310110 101012 Combine the following 1 2 3 4 5 6 1 2 3 4 5 6 4 5 6 1 2 3 3 2 1 6 5 4 How fast are the following algorithms BubbleSort Quicksort Mergesort Repeated Minimum sort Insertion sort Convert from binary into hexadecimal 11101011101001 1001011101110 101010011 1000000100001 Give addition and multiplication tables for Modulo 5 arithmetic Which of the following are propositions Eat at Joe s Make me smile I m in love with a big blue frog If cows were green then you couldn t see them in the grass Did you eat yet Money is the root of all evil rump96x CSI 2350 Review Dec 6 2004 10 Which of the following are true a If cows are green then pigs can y b If Xgt7 then X5gt12 c If 3 is a prime number then 10 is odd d If I won the lottery then I will go to Paris e If 9 is a prime number then 10 is bigger than 100 11 Given U l2345678910 Al23 B456 C678 give the following a A n B b A U B c B n C d A e B C 12 State and prove De Morgan s laws for sets 0011 0 Find the minimal Boolean Equations for the Give the commutative associative and distributive laws for Boolean Algebra Give the identity and complement laws for Boolean Algebra What is an equivalence relation What is a pa1tial order Ne gate a Vxx gt 7 b Vxx lt x l c EIxxlt3xgt7 Prove that xl3 x3 3x2 3xl truth tables lOll

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