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1. 1. 1 Name :………………………. THIRD SEMESTER B.TECH DEGREE EXAMINATION DISCRETE STRUCTURES ( S3 IT-CS) PREVIOUS YEARS’ QUESTIONS (2003 SCHEME) ES Note: 1. Repeated questions are merged and hence you may find decrease in number of questions for a particular year. RI 2.Only certain mistakes in the University question papers have been corrected PART A SE 2008 DECEMBER 1. Construct a truth table for the relation → ( → ). P 2. Define tautology wih example. 3. Convert into Predicate form TO a) Mark is poor but happy. b) Mark is neither rich or hapy. 4. Show that the Binary operation ‘*’ defined on (R,*), where x*y=max(x,y) is associative. 5. Prove that if any 14 integers from 1 to 25 are chosen, then one of them is multiple of another. SK ( ) 6. Define Abelian Group. 7. Prove the maximum number of edges in a simple graph with ‘n’ vertices is . 8. How many positive integer less than 10000 have exactly one digit equal to ‘9’ and have a sum of DE digits equal to 13? 9. Show that every subgroup of a cyclic group is normal. 10. Prove that every finite integral domain is a field. 2007 DECEMBER @ 1. For what truth values will the following statement is true. “It is not the case that houses are cold 2. Prove → ⇌ (∼ ∨ ) or haunted and it is false that cottages are warm or houses ugly.” a) ( ∧ ) ∨ b) ∨ (~ ⋁ ) 3. Write the dual of 4. Define Boolean and Pseudo Boolean Lattice. 5. How many edges are there in graph with 5 vertices each of edge 4 ? 6. Define planar graph. Give one example. http://at-desktop-series.tk A work of @ DESKTOP SERIES
2. 2. 2 7. For the semi group (NX) show that (TX) is a sub semi group. Where T is the set of multiples of a positive integer ‘m’. 9. In a ring show that – (− ) = (− )(− ) = 8. Show that every cyclic group of order n is isomorphic to (Zn +). 10. Show that every field is an integral domain. Explain integral domain field.(2006,2005,2004) 2007 REGULAR b)( → ) ∧ ( → ) a) ~(~ ∧ ~ ) 1. Construct a truth table corresponding to ES b) ( ) ( ) ∧ (∃ ) ( ) 2. If the universe is the set {a,b,c} eliminate the quantifiers a) (x) P (x) 3. Define Lattice, Give examples. RI 5. Are these monoid? ( ( ), )( ( ), ) where X is any given set. 4. Show that in a tree P= q+1. 6. If (G, *) is an abelian group then for all , ∈ (a*b)n = an *bn. SE 7. “Any Hamiltonian graph is two-connected”-Justify. ⊕( ∗ )= ⊕ b) ( ∗ ) ⊕ ( ∗ ) = 8. Prove the following Boolean identities a) 9. In any ring (R + .) show that zero element is unique. P 2006 DECEMBER TO a) ( ∨ ) ∨ b) [ ∨ ~( ∨ ~ )] 1. Write the dual of ( )= ( )= +, ( )( ) , ( )( ). 2. Let SK 3. Ina an office there are 13 clerks. Show that atleast two of them must have birth days during the same month. 4. Symbolize the predicate “x is the father of the mother of y”. 5. Define partially ordered set, give example. DE 6. Give an example of a graph with 6 vertices which is Eulerian. 7. Show that (B(x),0) is a monoid where B(x)set of all relations from X to X. ‘0’ is the composition of relation on B(X). 8. Check whether (R X) is a group. 9. The ring of even integers is a substring of ring of integers. @ 2006 REGULAR 1. Create a truth table for ( → ) ↔ (~ → ~ ) 2. A={1,2,3}.Obtain the power set P(A). 3. What so you mean by partially ordered set? Give any two examples. 5. Construct the tree of all the algebraic expression + ( − ) − ( − ( − )) 4. Is the poset A ={2,3,6,12,24,36,72} under the relation of divisibility a lattice. 6. Distinguish between tree and graph. 7. Complete the following table to obtain a semi group http://at-desktop-series.tk A work of @ DESKTOP SERIES
3. 3. 3 * a b c a c a b b a b c c a 8. Let a and b be the elements of a group then prove that (ab)-1 = b-1a-1. 9. Explain the principles of inclusion and exclusion. (2004) 10. Show that each element of a group has a unique inverse. ES 2005 DECEMBER Construct a truth table for ( ⇌ ) ⇌ ( ⋀ )⋁ ) 1. Explain well formed formula (WFF) with example. RI 2. If : → : → are Two functions show that 3. Define a predicate. Explain symbolically “If x is taller than y, then y is not taller than x”. 4. ONTO if g is ONTO. SE 5. Prove that if we select any 14 integers from 1 to 25 then one of them is a multiple of another. − + = , =, = 6. Is it possible to have a graph, “which is Eulerian but not Hamiltonian”. Why? √ 7. Solve 8. Show that (z7 ,t7 ,x 7) is a commutative ring with unity. P 9. State and prove the basic postulated of Boolean Algebra. TO 2005 REGULAR 1. Generate a truth table for the formula ( → ( → )) 2. Check whether the two statements are same. SK a) “Good food is not cheap” b) “Cheap food is not good” 3. Find all partitions of the set (a,b,c). 4. Define Pigeon hole principle. = 5. Define incident matrix, sub graph and spanning sub graph. DE () − + = 6. Find the sequence given by the generating function 7. Solve ∈ ={ / ∈ } 8. Let R be a Euclidian ring and A be an ideal of R. Show that there exists an element 9. Draw possible graphs, which are all connected (non-isomorphic) with four vertices. @ 10. Explain “Cyclic sub group“. 2004 DECEMBER 1. Obtain the equivalent formula using connections ∧ ~ →( ∨) ~ ⇄ 3. Prove that set relation − ( ⋃ ) = ( − )⋂( − ) 2. Let X= {1,2,3,4,5,6,7} and R= {(x,y) / (x-y) is divisible by 3}. Check whether R is equivalent. http://at-desktop-series.tk A work of @ DESKTOP SERIES
4. 4. 4 4. A flag is designed with 6 vertical stripes in yellow, green, red and blue. In how many ways can this be done o that not two adjacent stripes receive the same color? (+) 5. Obtain the discrete function corresponding to the generating formula = (−) 6. Show that the minimum number of edges in a simple graph with P vertices is (P ) 2 8. If a2 = a, show that each R is commutative for every element ∈ 7. Show that each element of a group has a unique inverse. 9. Explain switching function with example. ES 2004 REGULAR 1. In a group of 400 people, 250 can speak in English only and 70 can speak Hindi only. How many RI can speak both English and Hindi. 2. Describe the relation R if A={1,2,3,4} and B={1,4,6,8,9,} and aRb if and only if b=a2.Find the domain : → ,where R is the set of real numbers. If ( ) = ( )= + 3. Let : → − and range of R. SE find fog, gof, fof and gog. 4. Find the generating function of a sequence {ak} if ak = 2+3k. 5. Explain the vertex colouring problem. P 6. Draw all trees with five or fewer vertices. 7. Let A = {a,b} which of the following tables define a semigroup on A. Which define a monoid on A TO i) * a b ii) * a b SK a a a a b b b b b b a a 8. Find the distance between x and y DE a) x=110110 y=000101 b) x=001100 y=010110 9. Show that the permutation is even, while the permutation is odd. PART B @ MODULE I 1. a) Show that → can be derived from the premises → ( → ), ⋁ b)Prove that (∃ ) ( )⋀ ( ) ⇒ (∃ ) ( )⋀(∃ )⋀(∃ ) ( ) . 2. a) Show that the set R of Real Numbers is not a countable set. b)Explain Principle of Inclusion and Exclusion with an application (2008) ii) ( → ) → ⇒ ⋁ i) → ⇒ → ( ⋀ ) 3. a) Without using truth table show the implications b) Indicate the variables which are free and bound. Also show the scope of the quantifiers. http://at-desktop-series.tk A work of @ DESKTOP SERIES
5. 5. 5 i) ( )[ ( ) ∧ ( )] → ( ) ( ) ∧ ( ) ii)( )[ ( ) ⇌ ( )⋀(∃ ) ( )⋀ ( ) + 4. a) Show that [01] is uncountable. c) Show by using mathematical induction is divisible by 3. (2007) 5. a) From the following formula which are well formed? Give reason.Indicate which all are ii) (~ → ) → ( → ) i) → ( ∨ ) tautologies or contradiction b) Show that the set of real numbers is not countable. a) ~( ∧ ) → [~ ∨ (~ ∨ )] ⇔ ~ ⋁ b) ( ∨ ) ∧ [~ ∧ (~ ∧ )] ⇔ ~ ⋀ 6. Without using truth table show that following logical equivalence ES a) ( , ) ( , ) if and only if xv=yu on the set of ordered pairs of integers. 7. From the following which are equivalence relation? Give reason. − b) .(2007) RI 8. a) Show that set of ordered pairs NxN is countable. b) Show by using mathematical induction 2n < n! for n ≥ 4. i)~( ⇌ ) ⇔ ( ⋁ ) ⋀ ~( ⋀ ) ii) → ( ∨ ) ⇔ ( → ) ∨ ( → ) 9. a) Show the following equivalences b) Show that ( ∨ ) ∧ ~[~ ∧ (~ ∨ ~ )] ∨ [~ ∧ ~ ) ∨ (~ ∧ ~ )] is a tautology. (2006) SE 10. a) State and explain pigeon hole principle. b) Show that any 11 numbers chosen from the set {1,2,4…….20}, will be a multiple of another. ( ∪ )=( )( ) c) Prove by mathematical induction that if a set A has n elements then P (A) has 2n elements. P ∩ 11. a) Let A,B and C be subsets o U.Prove that ∩ are finite and | ∪ | = | | + | | − c) If A and B are finite sets then prove that ∪ TO b) Show that if R1 and R2 are equivalent relations on A, then is equivalent relation on A. | ∪ |.(2006) 12. a)”If wages increase then there will be inflammation. The cost of living will not increase if there is SK no inflammation. Wages will increase therefore the cost of living will increase” . Test the validity of c) Use mathematical induction to prove the = is divisible by 3 whenever x is a positive the argument. a)Using the principle of inclusion-exclusion count the number of integral solutions to + + integer. DE =, ≤ ≤;≤ ≤ ;− ≤ ≤ 13. b)Explain Canto’s theorem of pair sets and give examples for reflexive, symmetric and transitive a)Prove without using truth table → ( ⋁ ) ⇔ ( → ) ∨ ( → ) relations. b) Show that ( )( ( ) → ( )) ∩ ( / ) → ( )))) ⟹ ( ( ) → ( )) 14. @ 15. a)Test the validity of the arguments: I can graduate only if I have a GPA o 3.5. Either I am smart or b) Prove that → ( ∨ ) ⇋ ( ⋀~ ) → I do not have a GPA of 3.5,I did not graduate. Hence I am not smart. a) Show that (∃ )( ( )⋀ ( ) ⇒ (∃ ) ( )⋀(∃ )( ( )) 16. ={ , / ∈ } c) Let R and S be two relations on the set of positive integers I: = {( , )/ ∈ } Find R.S, R.R , R.R.R , and R.S.R http://at-desktop-series.tk A work of @ DESKTOP SERIES
6. 6. 6 17. a) ( ⋃ ) − = ( − )⋂( − ) b) Let R be a binary relation on the set of all positive integers such that R={(a,b)/a-b} is an odd posistive integer. Is R Reflexive, Symmmetric,Anitsymmetric or Transitive? c) Using mathematical induction show that for all positive integers ∩ are finite and ( ∪ ) = ( ) + 18. a) If A and B are finite sets, then prove that ∪ 1.2.3 + 2.3.4 + ……….+(n(n+1)(n+2))=n(n+1)(n+2) ( )− ( ∩ ) ={ : ∈ , = } b) Find the cardinal number of each set. = { , , , ,….} i) ES = { , , , ,…..} ii) iii) c) If R be a relation in the set of integers defined by R = {(x,y); xєz, yєz , (x-y) is multiple of 3}. RI Show that it is an equivalence relation. What is the equivalence class of 0? How many equivalence classes are ther? MODULE II SE 1. a) Prove that a connected graph G is Eulerian iff it can be decomposed into circuits. b)Identify the sequence having the expression as a generating function . 2. a) Show that a linearly ordered Poset is a distributive Lattice. P b)Explain how digraphs are represented in computer. 3. a) In a lattice show that if a ≤b and c ≤ d then a*c ≤ b*d. TO b) Prove that it is impossible to have a group of a people at a party, such that each one knows = + +. … … + () exactly 5 of the others in the group. (2007) 4. a) Prove that the generating function of the sequence if A(x) is the SK generating function of the sequence a0, a1, …………an. (,20072005) b) Show that in a graph number of vertices of odd degree is even. 5. a) Prove that if G is a connected graph, G is Eulerian if and only if every vertex is of even degree.(2004) DE b) Show that K5 is not planar. 6. a) Show that a lattice with three or fewer elements is a chain. b) Can there exists a simple graph with 15 vertices each of degree 5. 7. a) Show that a graph G can be 4 coloured if G has a Hamiltonian cycle. b) Show that K33 is not planar. (2006) @ () 8. a) Let A(x) be the generating function of the sequence a0, a1, …………an. Show that the generating = − function of the sequence bn= a0+ a1+ …………an is b) Solve the recurrence relation , d1=1 and d2=7. 9. a) Write down a homogeneous linear recurrence relation of degree k.Also write its characteristic polynomial equation. b) Explain backtracking technique for solving a recurrence relation. (2006) 10. a) Show that a connected graph G is an Euler graph if it can be decomposed into circuits. b) Explain adjacency matrix and find it for the given graph. http://at-desktop-series.tk A work of @ DESKTOP SERIES
7. 7. 7 d) Solve the recurrence relation + = ≥ = =− . ES 11. a) Define Poset. Explain how Poset become a Lattice? 12. Draw a graph with 7 vertices which is Eulerian but not having Hamiltonian circuit. = + ≥ , there are C1 and C2 such 13. a) Show that a graph G can be 4-coulers if G has Hamiltonian cycle. RI = + d) Show that Fn is the Fibonacci relation, ) √ that where C1 and C2 are evaluated for the initial condition. SE 14. a) Determine all minimal chains with end points 1and n in [Dn : 1], where n= with P1 and P2 = − = =. distinct prime numbers and K1 and K2 are positive integers. − + = = =. b) Solve the recurrence relation 15. Solve by method of generating function b) Let G = {(S,A),(a,b),S,φ} where φ consists of the production → ,→ ,→ ,→ P 16. a) If G is a connected planar graph with r regions, then prove that v+r-e=2 with usual notations. ,→ c) Solve the recurrence relation = , ≥ with the boundary condition = using TO .Prove that the word abab is ambiguous. generating function. 17. a) Draw the undirected Graph G corresponding to adjacency matrix: SK = = { , , }, ={ , } DE b) Determine the type of the grammar G which consists of the production {→ ,→,→} = ,→ i) = { , , }, ={ , , } ) ={ → ,→ , → , − , → , → , →} @ − + = e) Find the general solution to the non linear recurrence relation, 18. a) Explain the use of preprocessor directives for conditional compilation. b)Denote minimum (a,b,c) and maximum (a,b,c,d) using only ternary operators in C. 19. a) What are bit-level operators in C? Explain. b) Write a program to compute A+B*C where A,B and C are polynomials. 20. What are the storage classes in C? Illustrate the use of each of them. Compare them on the basis of scope, life time and initialization. http://at-desktop-series.tk A work of @ DESKTOP SERIES
8. 8. 8 MODULE III 1. a)Prove that + is irreducible over the field F of integers mod 11. b)If G is a group and ( ) = , for all n ≥ 1, show that = / ∈ is a Normal subgroup of G. b)If = + √ ∶ , ∈ , ( , +, . ) is an integral domain. 2. a) Define Euclidean domains and integral domains with example. b) Show that ( ⊕⊙) is a ring where 3. a) State and prove Lagrange’s theorem. ⊕=+− ⊙=+− ; ,∈ ES 4. a) Let (B,*,+,’,0,1) be a Boolean algebra. Define + and . on the elements of B as a+b=(a*b1)(a1*b) and a.b=a*b e) Let (G,*) be a group a є G let : →is defined by ( ) =a*x*a-1 for every x є G.Prove that f is RI Show that (B,+,.,1) is a Boolean ring with identity 1. a) Show that a subset ≠ of G is a subgroup of (G,*) if and only if for any pair of elements a,b є an isomorphism from G on to G. (2007) 5. SE s, a*b-1 є s. b) Show that (z7,+,x) is a commutative ring with identity. 6. a) Show that I a group (G,*) is of even order then there must be an element aє G where a≠e such that a*a = e. P b) Prove that every finite integral domain is a field. (2007,2006) 7. a) Show that every finite group of order n is isomorphic to a permutation group of order n. TO b) A code can detect all combination of K or fewer errors if and only if the minimum distance between any two code words is atleast (K+1). 8. a) Simplify the Boolean expression SK i) (a*b+(‘(a+b)’ ii) (a*c)+c+((b+b)’*e) a) Construct a truth table for the Boolean function : → determined by the polynomial b) Define ring homomorphism, give example.(2006) ( )=( )∨ , , ∧ ∨ ∧ 9. (2006) DE b) Define Boolean algebra. Write down the axioms of Boolean algebra. What are the main difference between Boolean algebra and algebra of real numbers. (2006,2004) c) Show that if ‘f’ is a homomorphism from a commutative semigroup (S,*) on to a semigroup 10. a) Let (S,*) and (T,*) be monoids with identities e and e1, respectively. Let ∷ → be an (T,*), then (T,*) is also commutative. @ b)Let R be a Euclidian ring and A be an ideal of Reshow that there exists an element ∈ isomorphism, thn prove that f(e)=e1 = { / ∈ }. (2006) 11. a)Define the operation ‘*’ on the set Q of rationals by a*b=a+b-ab. Show that (Q,*) is a semigroup with identify. 12. a)Show that − is irreducible over F, the field of integer mod 11. b)Explain congruence relation and Eucledian domain. = + =. b)Prove using the rule of Boolean Algebra http://at-desktop-series.tk A work of @ DESKTOP SERIES
9. 9. 9 13. a)Describe the quotient semi-group for S=Z with ordinary addition and relation R defined as aRb if b)Show that in a group (G,8).If for every , ∈ , ( ∗ ) = ∗ . Then (G,*) must be abelian. a≡b(mod3). 14. a) Show that the algebraic system| (C,+,’) where C is the set of complex numbers and + and ‘ 15. b) If a Boolean Algebra and ∈ is a minimal if a≠0 and if , for every xєB, x≤a or x=0.Show that a denote complex addition and multiplication in a field. is an atom iff a is minimal. b)Minimize the switching function ∑ ( , , , , , ) 16. a)If U is an ideal of a ring, R and I є U, prove that U=R. ES 17. a) Let S=QxQ, the set of ordered pairs of irrotational numbers, with the operation * defined by (a,b)*(x,y)=(az,ay+b) i) Find (3,4)*(1,2) and (-1,3)*(5,2) RI ii) Is S a semigroup? Is it commutative? iii) Find the identity element of S. ={ , , , , } b) Consider (Z,*) where *is defined by a*b=a+b-ab. Show that (Z,*) is a group or monoid. SE 18. a) Draw the transition diagram for the is as in the table a b . . . q0 q0 q1 P . . . q1 q0 q2 TO . . . q2 q2 q2 SK Also check whether the strings aaab and bbaa are acceptable to M. d) Define a group homomorphism from (G,*) to (G’,*) and prove that every cyclic group of order n is isomorphic to (Zn,tn) -------End------- DE @ http://at-desktop-series.tk A work of @ DESKTOP SERIES