MODERN APPLIED ALGEBRA

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1 Paper IV (Elective -) - Curriculum ACHARYA NAGARJUNA UNIVERSITY CURRICULUM - B.A / B.Sc MATHEMATICS - PAPER - IV (ELECTIVE - ) MODERN APPLIED ALGEBRA 90 hrs (3hrs / week) UNIT - (30 Hours). SETS AND FUNCTIONS : Sets and Subsets, Boolean algebra of sets, Functions, Inverses, Functions on S to S, Sums, Products and Powers, Peano axioms and Finite induction.. BINARY RELATIONS : Introduction, Relation Matrices, Algebra of relations, Partial orderings, Equivalence relations and Partitions, Modular numbers, Morphisms, Cyclic unary algebras. UNIT - (0 Hours) 3. GRAPH THEY : Introduction - Definition of a Graph, Simple Graph, Konigsberg bridge problem, Utilities problem, Finite and Infinite graphs, Regular graph, Matrix representation of graphs - Adacency matrix, Incidence matrix and examples; Paths and Circuits - Isomorphism, Sub graphs, Walk, Path, Circuit, Connected graph, Euler line and Euler graph; Operations on graphs - Union of two graphs, Intersecton of two graphs and ring sum of two graphs; Hamiltonian circuit, Hamiltonian path, Complete graph, Traveling salesmen problem. Trees and fundamental circuits, cutsets. UNIT - 3 (5 Hours) 4. FINITE STATE MACHINES : Introduction, Binary devices and states, Finite state machines, State diagrams and State tables of machines; Covering and Equivalence, Equivalent states, Minimization procedure. 5. PROGRAMMING LANGUAGES : Introduction, Arithmetic expressions, Identifiers, Assignment statements, Arrays, For statements, Block strutures in ALGOL, The ALGOL grammar. UNIT - 4 (5 Hours) 6. BOOLEAN ALGEBRAS : Introduction, Order, Boolean polynomials, Block diagrams for gating networks, Connections with logic, Logical capabilities of ALGOL, Boolean applications. Prescribed Text Book : Modern applied Algebra by Dr. A. Ananeyulu, Deepti publications, Tenali. Reference Books :. Modern applied Algebra by Garrett Birkhoff and Thomas C.Bartee, CBS Publishers and Distributors, Delhi.. Graph Theory with applications to Engineering and Computer Science by Narsingh Deo, Prentice-Hall of India Pvt. Ltd., New Delhi.

2 ACHARYA NAGARJUNA UNIVERSITY CURRICULUM - B.A / B.Sc MATHEMATICS - PAPER - IV (ELECTIVE - ) MODERN APPLIED ALGEBRA QUESTION BANK F PRACTICALS Paper IV (Elective -) - Curriculum UNIT - (SETS AND FUNCTIONS, BINARY RELATIONS). i) Is the cancellation law A B = A C B = C true? If not give example? ii) Is the cancellation law A B = A C B = C true? If not give example?. Prove that S S ( S T)and that S S ( S T)and S = S ( S T). 3. If ABC,, are three sets, prove that ( A B) C = ( A C) ( B C). 4. Find a necessary and sufficient condition for S + T = S T where S + T = ( S T ) ( S T). 5. Give an example of a function which is a surection but not inection. 6. Show that the Peano s successor function is an inection but not surection. 7. Find the number of functions from a finite set S of n elements to itself. Among these i) how many are surections ii) how many are inections? 8. Show that the functions f ( x) = x / and gx ( ) = x for x Rare inverses of one another. 9. Prove by induction in Pm, + r= m+ s r= s. 0. If f, f,..., f m are inections then show that fm o fm o... o f is an inection.. Prove by induction that 3 n + n is divisible by 3 for all n. n nn ( + ). Prove by induction that k = where k = n is any positive integer. 3. Let X = { a, b} and Y = {, c d,}. e Write down the tabular representation for the relation α on X and Y defined by the list : aα c, aα d, aα e, bα c, bα d, bα e. 4. Find the matrix of the relation α on X = { a, b}and Y = {, c d,}which e is defined by the list : aα c, aα d, aα e, bα c, bα d, bα e. 5. Give an example of a relation which is neither reflexive nor irreflexive. Also give its graphical representation and relation matrix. 6. If ρ is symmetric, prove that ρ ρ n... ρ is symmetric. 7. Show that a finite poset has a least element iff it has exactly one minimal element. 8. If ρ is reflexive and transitive then show that ρ ρ is an equivalence relation on a set S. 9. Let A= ( S, f )be a finite unary algebra with k elements. Define arb in A to mean that for some n n N, f ( a) = b. Show that R is reflexive and transitive. 0. Let A= [ S, f ] be any finite unary algebra with k elements. Define arb in A to mean that for some n n N, f ( a) = b. Show that A is cyclic iff for some a S, arb for all b S.

3 3 Paper IV (Elective -) - Curriculum UNIT - (GRAPH THEY). Draw the graph GV (, E)where V = { A, B, C, D, E}, E = { AB, AC, BC, DE}.. Find the Edge set E of the graph G = ( V, E)given by B A C E D 3. Explain Konigsberg bridge problem and draw its graph. 4. Explain three-utilities problem and draw its graph. 5. Draw a graph with the adacency matrix Draw a graph with the adacency matrix Draw a graph with the incidence matrix 8. Find the adacency matrix of the graph given by e e 5 e 7 e 8 e v e3 e 4 v e 6 9. Find the adacency matrix of the graph given by D C A B

4 4 30. Find the incidence matrix of the graph given by e 8 e 7 Paper IV (Elective -) - Curriculum e 9 e e 4 3. Find the union, intersection and ring sum of the following two graphs. v d v a b c f G e v e 5 e 3 e v v 5 G v e 6 a g k v c h v 6 l v 5 3. Show that the two graphs are isomorphic. 33. Show that the following graphs are not isomorphic to each other. v v u u v 5 v 6 u 5 u 6 v 8 v 7 u 8 u 7 u 4 u Draw a circuit from the following graph which is of length nine. v v 5 v 6 v v 0 v 7 v 9 v Draw all the circuits of the following graph. v e e 5 e e 3 e 4 e 6 v e 7

5 36. List all paths from v to v 8 in the following graph. 5 Paper IV (Elective -) - Curriculum e e 4 v e v e 5 e 6 e 3 v 7 e 7 v 5 v 6 e e 0 e 9 e 3 e 8 e v Find the eccentricity and centre of the graphs. (i) a (ii) e a d b c 38. Find the rank and nuclity of the spanning tree. f d b c v b v c 3 c c b v 5 b 6 b 3 c 4 b 5 c 6 v Draw all trees of three labeled vertices. 40. Find the edge connectivity of the complete graph of n -vertices. c 5 v 7 UNIT - 3 (FINITE STATE MACHINES, PROGRAMMING LANGUAGES) 4. Find the output string when the input string 0 is run into the following machine. Present v ξ state 0 0 s s Give the state diagram of the following transition table. Present Next state Output state

6 43. Draw the state diagram for the following machine 6 Paper IV (Elective -) - Curriculum Present Next state Output state 0 0 s 0 s 0 0 s s 5 s Write the state table of the following machine.,,,, Establish a morphism from M and M given below : ν ξ M ν ξ M a b c 0 0 b a c 0 0 c c a Minimise the number of states in the following machine. Present Next state Output state a 0 a a 0 a Minimise the number of states in the following machine. 0, 0, s 0 s, 0 s 3, 0, 0, 0, 0

7 48. Minimise the number of states in the following machine. Present Next state Output state a 0 a a 0 a Write the ALGOL expressions for the following Mathematical expressions. 3 3 i) a b 3ab( a+ b) ii) AB + C D E iii) π ( r p) 50. Write the ALGOL expressions for the following Mathematical expressions. 7 Paper IV (Elective -) - Curriculum n r iv) P + v) 00 log x y + xy / y i) sin e x y ii) sin 3 x + iii) e A sin A iv) b+ b 4 ac a v) + sin x 5. Convert the following ALGOL expressions to conventional mathematical expressions. i) A B C ii) A ( B C) iii) A B C iv) A/ B C D v) A B C D Evaluate the above for A=, B= 3, C= 4, D= Write the mathematical expression for the following ALGOL expressions i) sqrt( s ( s a) ( s b) ( s c)) ii) sqrt (exp(sin A) cos( A 5)) iii) b+ sqrt( b 4 a c) / a iv) (exp( x) x+ exp( / x)/(exp( x ) + sqrt( x)) v) sqrt (exp( A) sin( A))/ x 53. Write down the effect of the following for statements. i) for i:= step until 0 do S ; ii) for i:= 4 step until 7 do S ; iii) for x:= 0 step 0 until do S ; iv) for x:= step 0 until 05 do S ; v) for x:= 5 step until 4 do S ; 54. Write the effect of executing the assignment statements of the following ALGOL block. begin real abc,, ; c: = 5; end a : = 4 ; b: = a+ 7; c: = 3 a b; 55. Write the ALGOL program which generates an array K with Ki [] = i! for i =,,..., Write ALGOL program to compute the mean of 0 observations. 57. Write an ALGOL program for finding the area of a triangle, given its three sides.

8 8 Paper IV (Elective -) - Curriculum 58. Two one-dimensional arrays X and Y each contain 50 elements. Write the ALGOL program to compute LX = 50 Σ = X (Length of the vector X ), LY = 50 Σ Y = (Length of the vector Y ) INPROD = 50 Σ = XY (Inner product of X and Y ) a a a 59. Write ALGOL program to multiply the matrix A = a a a a a a Write down the ALGOL block for finding the roots of ax + bx + c = 0 ( a 0) b by a column matrix B = b b3,. UNIT - 4 (BOOLEAN ALGEBRAS) 6. What is two element Boolean algebra. 6. In a Boolean algebra show that x z x ( y z) = ( x y) z. 63. In a Boolean algebra prove that ( x y ) ( x y) = ( x y) ( x y ). 64. In a Boolean algebra prove that ( x y) ( y z) ( z x) = ( x y) ( y z) ( z x). 65. Let ab, B, a Boolean algebra. If is denoted by + then prove that a+ b is an upper bound for the set { ab, } and also a+ b=sup{ a,}. b 66. Given the interval [ ab, ] of a Boolean algebra A. Show that the algebraic system [[ ab, ],,,, ab. ] is a Boolean algebra, where x = ( a x ) b, x [ a, b]. 67. Draw the block diagram of p ( p q). 68. Draw the block diagram of ( A A ) ( A A ) 69. Draw the block diagram of ABC A B C A B C. 70. Write the gating network representing the Boolean expression ( x y) ( x y z ) ( y z). 7. Write the gating network representing the Boolean expression [( x x ) x ] ( x x ). 7. Write the Boolean expression for the gating network. x y 3 z 73. Show that [( p q) ( p r)] ( p r) is a tautology. 74. Show that ( p q) [( r p) ( r q)] is tautology, regardless of r.

9 75. Show that ( p p) ( p p ) is an absurdity. 9 Paper IV (Elective -) - Curriculum 76. Construct the truth table and write the logic diagram for the following Boolean polynomial, Pxyz (,, ) = ( x y) ( y z ). 77. Construct the truth table and write the logic diagram for the following Boolean polynomial, P( x, y, z) = ( x z) ( x y) ( y z) x 78. Write an ALGOL program to compute F x 7 3 ( x+ ) x < 5 ( ) = for 9 4 /( + x ) x 5 x ranging from 0 to 0 in steps of 0. x 79. Write an ALGOL program to compute F x 35 x if x < 5 ( ) = for ( x 5)/( + x ) if x 5 x ranging from to 0 in steps of Write the ALGOL program which computes the relation matrix for the relation ρ where ρ is a binary relation on a set X = { x, x,..., x n }.

10 0 Paper IV (Elective -) - Curriculum ACHARYA NAGARJUNA UNIVERSITY B.A / B.Sc. DEGREE EXAMINATION, THEY MODEL PAPER (Examination at the end of third year, for 00-0 and onwards) MATHEMATICS PAPER - IV (ELECTIVE - ) MODERN APPLIED ALGEBRA Time : 3 Hours Max. Marks : 00. State Peano axioms. SECTION - A (6 X 6 = 36 Marks) Answer any SIX questions. Each question carries 6 marks. Show that the relation m< n means mn (meaning that m is a divisor of n) is a partial ordering of the set of all positive integers. 3. Explain utilities problem. 4. If a graph (connected or disconnected) has exactly two vertices of odd degree, prove that there must be a path oining these two vertices. 5. Draw the state diagram for the following machine. Present v ξ state Write ALGOL expressions for i) b + b 4ac a x ii) sin 3 iii) a b + c q y 4 d 7. Define Boolean algebra. 8. Prove that in any Boolean algebra, a x =0 and a x = imply x = a. 9.(a) SECTION - B (4 X 6 = 64 Marks) Answer ALL questions. Each question carrie6 marks Prove that a function is left invertible iff it is one one. n (b) Prove by induction that Σ k nn ( + )( n+ ) = where n is any positive integer. k = 6 0.(a) Prove that an equivalence relation on a set S gives rise to a partition on S. (b).(a) If ρ and σ are reflexive and symmetric relations on a set S, then show that the following are equivalent. i) ρσ is symmetric ii) ρσ = σ ρ iii) ρσ = σ ρ. Prove that a connected graph G is an Euler graph iff it can be decomposed into circuits.

11 Paper IV (Elective -) - Curriculum (b) Find the aacency matrix and incidence matrix of the graph given by e e 5 e 7 e 8 v e 4 e e3 e 6 v.(a) (b) Prove that the number of vertices of odd degree in a graph is always even. Draw all the circuits of the following graph. v e e 5 e e 3 e 4 e 6 v e 7 3.(a) (b) 4.(a) (b) Prove that the relation of equivalence of machines is an equivalence relation. Minimize the number of states in the following machine. Present Next state Output state a 0 a a 0 a Explain i) for statement ii) BLOCK structures in ALGOL. Two one-dimensional arrays X and Y each contain 50 elements. Write the ALGOL program to compute LX = 50 Σ = X (Length of the vector X), LY = 50 Σ Y = (Length of the vector Y), INPROD = 50 Σ = XY (Inner product of X and Y)

12 Paper IV (Elective -) - Curriculum 5.(a) In any Boolean algebra, if a x = a yand a x = a y, then prove that x = y. ' (b) Write the gating network representing the boolean expression [ ] ( x x) x3 ( x x). 6.(a) If any Boolean algebra B = [ A,,, '], prove that the relation a< b is a partial ordering of A. Moreover, in terms of this partial ordering, prove that a b = glb { ab, } and a b = lub { ab, }. (b) Show that ( p q) [( r p) ( r q)] is a tautology, regardless of r. Y

13 3 Paper IV (Elective -) - Curriculum AACHARYA NAGARJUNA UNIVERSITY B.A / B.Sc. DEGREE EXAMINATION, PRACTICAL MODEL PAPER (Practical examination at the end of third year, for 00-0 and onwards) MATHEMATICS PAPER - IV (ELECTIVE - ) MODERN APPLIED ALGEBRA Time : 3 Hours Max. Marks : 30 Answer ALL questions. Each question carries 7 marks. 4 7 = 30 M (a) If f, f,..., f m are inections then show that fm o fm o... o f is an inection. (b) If ρ is reflexive and transitive then show that ρ ρ is an equivalence relation on a set S. (a) Draw a graph with the adacency matrix (b) Show that the two graphs are isomorphic. 3 (a) Give the state diagram of the following transition table. Present Next state Output state (b) Convert the following ALGOL expressions to conventional mathematical expressions. i) A B C ii) A ( B C) iii) A B C iv) A/ B C D v) A B C D Evaluate the above for A=, B= 3, C= 4, D= 5 4 (a) Draw the block diagram of ABC ABC ABC. (b) Show that ( p p) ( p p ) is an absurdity. Written exam : 30 Marks For record : 0 Marks For viva-voce : 0 Marks Total marks : 50 Marks