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# DISCRETE STRUCTURES CS 381

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This 3 page Class Notes was uploaded by Armani Kunde on Monday September 28, 2015. The Class Notes belongs to CS 381 at Old Dominion University taught by Khaled Ibrahim in Fall. Since its upload, it has received 28 views. For similar materials see /class/215311/cs-381-old-dominion-university in ComputerScienence at Old Dominion University.

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Date Created: 09/28/15

CS 381 Midterm Review You should be able to Nt I 5 05quot b ib ib Ib Il WOON ALANHOquot39 Produce the truth table for a proposition Prove a proposition is a tautology contradiction or is logically equivalent to another proposition using the logical equivalence laws and using truth tables Translate quanti ed predicates to English and vice versa Given the appropriate de nitions for variables and a universe of discourse determine the truth value of a statement that contains quanti ers Given a set of premises construct a proof using the rules of inference Given de nitions for multiple sets determine subset superset strict subset and strict superset relationships Determine set union intersection and difference Determine the power set of a set Determine the Cartesian product of two sets Determine the cardinality of a set including the cardinality of a power set Draw a Venn diagram for sets Apply a function to a set Given a function determine if it is inj ective surjective andor bij ective Given two function de nitions nd the composition of the functions Compute the ceiling and oor of any real number Write pseudocode to solve a listbased problem Given pseudocode for an algorithm write a function that calculates the number of times a particular line of the algorithm is executed Given a function determine and prove bigO bigOmega and bigTheta bounds Determine whether two integers are congruent mod X where X is any integer Compute the prime factorization of some number less than 100 Find the least common multiple of two natural numbers Find the greatest common divisor of two natural numbers Review Problems 1 N E 4 Produce the truth table for the following propositions a XAy gtzvy b a lt gt b gt c c XA yAsz Prove each of the following first using logical equivalences then again using truth tables a a lt gt b gt c is logically equivalent to a A b A c v a A b b X A y v 2 v X v y A z is atautology c a gt b A a A b is a contradiction Translate quanti ed predicates to English and vice versa Examples include the first 4 questions from quiz 2 and the retake of quiz 2 Given the appropriate de nitions for variables and a universe of discourse determine the truth value of a statement that contains quantifiers X 3lollipops pies danishes y 3apple cherry grape bananna TXy X are available in y avor Assume that Lollipops are available in apple cherry grape and bananna avors Pies and danishes are available only in apple and cherry avors Which of the following statements are true a VX Vy TXy b VX Vy TXy c VX Vy TXy 3 1 EIX Ely TXy e EX Ely TXy f El X Ey TXy Ely EX TXy P QO Vy 3X TXy i Ely VX TXy 5 Given a set of premises construct a proof using the rules of inference See quiz 2 and quiz 2 retake for other examples a Premise a b v a Premise b gt c Premise e lt gt c Prove that a gt e Let A 0 3 5 9 and Bl l 3 6 911 6 a What is A UB b What is A n B c Is A 7 B D BA d IngBnA 7 What is the power set of A 8 WhatisAgtlt B WhatisB gtlt Agtlt B 9 What is lAl What is the cardinality ofthe power set of B 10 Draw a Venn diagram that includes A B and A X B Let fX x2 3 Let gX x 71 11 What is gA What is fB 12 Is fX injective surjective andor bijective What about gX 13 Let h f0 g What is hA 14 What is l f05 What is mom 15 Write pseudocode to nd the nexttosmallest integer in a nonordered length N list of integers 16 Given the pseudocode below write a function that calculates the number of times line 3 would execute lforilto Nl 2 forjltoN5 3 aiail

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