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Fundamentals: 1.Draw venn diagram showing: (i) AB AC but BC, (ii) AB AC but BC. 2. Let U={1,2,3,4,5,6,7,8,9},A={1,2,4,6,8} B={2,4,5,9},C={x|x is a positive integer and 𝑥 ∗ 𝑥< 16} and D={7,8} Compute a. (AB)-C b. A-B c. A symmetric difference B. d. C-D. Show it using venn diagram. 3.In a class of 80 students,50 students know English,55 know French and 46 know know German language. 37 students know English and French,28 students know French and German,7 students none of the languages.Find out a. How many students know all the three languages? b. How many students know exactly 2 languages? c. How many know only one language? 4. Among integers 1 to 300,how many of them are divisible neither by 3,nor by 5,nor by 7? How many of them are divisible by 3 but not by 5,nor by 7? 5.Define characteristics function. Logic: 1. Prove by mathematical Induction : a. a+ar+a𝑟 2 +........+a𝑟 𝑛−1=a(1-𝑟 𝑛 )/(1-r) b. 1+2𝑛 <3𝑛 for n>=2. 2. Make a truth table for the following: (i) (pq) r (ii) (pq) r (iii) (p q) (p r) 3. State and prove De Morgan’s law for logic. 4. Is ((pq) (pq)) q a tautology 5. Prove the following: (i) p (p q) (pq), (ii) p (pq) (p q) 6. Consider the following conditional statement: If the flood destroy my house or the fires destroy my house, then my insurance company will pay me. Write the converse, inverse and contrapositive of the statement. 7. Given the following statements as premises, all referring to an arbitrary meal: If he takes coffee, he does not drink milk. He eats crackers only if he drinks milk. He does not take soup unless he eats crackers. At noon today, he had coffee. Whether he took soup at noon today? If so what is the correct conclusion. 8. There are two restaurants next to each other. One has a sign says “Good food is not cheap” and other has a sign that says “Cheap food is not good”. Are the signs saying the same thing? 9. Is the following argument valid? If taxes are lowered, then income rise Income rise Taxes are lowered 10. Write the following statement in symbolic form using quantifiers: (i) All students have taken a course in mathematics. (ii) Some students are intelligent, but not hardworking. 11. Let p(x): x is mammal and q (x): x is animal. Translate the following in English: (x)(q(x) (p(x)) ) 12. Let A = {1, 2, 3, 4, 5}, determine the truth value of the following: (i) (xA)(x + 3 = 10), (ii) (xA)(x + 3 <5). 13. Write the negation of the following statement: x R x > 3 x2 > 9 14. Prove the following or provide a counter example: AB AB A = B 15. Prove or disprove the statement that if x and y are real numbers: (x2 = y2) (x = y). 16. Let n be an integer, prove that n2 is an odd then n is odd. 17. Use the method of contradiction to prove that 5 is not a rational number. Counting: 1. Suppose a valid password consists of seven characters,the first of which is a letter chosen from the set {A,B,C,D,E,F,G} and the remaining six characters are letters chosen from the English alphabet or a digit. How many passwords are possible. 2.A microcomputer manufacture who is designing an advertising compaign is considering six magzines,three newspapers,two television stations,and four radio stations. In how many ways can six advertisements be run if a. all six are to be in magazines. b. Two are to be in magazines,two are to be in newspapers,one is to be on television,and one is to be on radio. c. In how many ways can a committee of three faculty members and two students be selected from seven faculty members and eight students. 3.Show that if seven colors are used to paint 50 bicycles,atleast eight bicycles will be the same color. 4.Consider a regular hexagon,whose sides are of length 1 unit. Show that if any seven points are chosen in this region,then two of them must be no further apart than 1 unit. 5. Show that if 30 dictionaries in a library contain a total of 30,000 pages,then one of the dictionaries must have atleast 1001 pages. 6.Solve the recurrence relation 𝑏𝑛 =7𝑏𝑛−1 -12𝑏𝑛−2 ,𝑏1 =1 7. Solve 𝑓𝑛 =𝑓𝑛−1 + 𝑓𝑛−2 , 𝑓1 =𝑓2 =1. Relations and Digraphs 1.Let A={1,2,3,4},B={1,4,6,8,9} Find the relation R such that b=𝑎2 and Construct the diagraph. 2. what is the path in the diagraph. Let A={1,2,3,4} R={(1,1),(2,2),(1,2),(1,3),(2,3),(2,4),(3,4),(4,4)} Find the diagraph of Relation R. Find the paths of length 1,2,3 and 4. 3. Find M𝑅 3 of the above relation. 4. Proof 𝑅 2 =𝑀𝑅 2 5. Let A={1,2,3,4) Determine whether the relation is reflexive,irreflexive,symmetric,asymmetric,antisymmetric,or transitive. a. R={(1,1).(2,2),(3,3),(1,2),(2,1),(3,4),(4,3),(4,4) b.R=AxA c. R is less than d. R is greater e. R is equal to. 6. Define equivalence relation. Prove that the Relation divide by(\) is equivalence relation or not. Relation is taken over set of integers. 7.Let R and S are relations over set AxA,where A={1,2,3,4} R={(1,1),(2,2).(3,3),(1,3),(2,3)} S={(4,4),(3,2),(4,4),(4,1)} Find a. 𝑅̅ b. R∪ 𝑆 c. R∩ 𝑆 d. 𝑆 −1 5.3. Let A = {1, 2, 3, 4}, and R is a relation defined by “a divides b”. Write R as a set of ordered pair, draw directed graph. Also find R-1 4. Let R be a binary relation defined as R = {(a, b) R2: a-b < 3}, determine whether R is reflexive, symmetric and transitive. 5. Let A = {1, 2, 3, 4, 5, 6}, construct pictorial description of the relation R on A defined as R = {(a, b): (a-b)2 A}. 6. Let A = {1, 2, 3, 4}, give an example of a mapping which is (i) neither symmetric nor antisymmetric, (ii) anti-symmetric and reflexive but not transitive, (iii) transitive and reflexive but nit anti-symmetric. 7. Let R be a relation on the set A = {a, b, c} defined by R = {(a, b), (b, c), (d, c), (d, a), (a, d), (d, d)}. Write the relation matrix of R and find (i) reflexive closure of R, (ii) symmetric closure of R and (iii) transitive closure of R. 8. Show that a relation R defined on the set of real numbers as (a, b) R (c, d) iff a2 + b2 = c2 + d2. Show that R is an equivalence relation.