III BA, B.Sc. Mathematics Paper IV (Elective-2) - Acharya Nagarjuna ...

III B.A./B.Sc. Mathematics1Paper IV (Elective -2) - CurriculumACHARYA NAGARJUNA UNIVERSITYCURRICULUM - B.A / B.ScMATHEMATICS - PAPER - IV (ELECTIVE - 2)MODERN APPLIED ALGEBRA90 hrs(3hrs / week)UNIT - 1 (30 Hours)1. SETS AND FUNCTIONS :Sets and Subsets, Boolean algebra of sets, Functions, Inverses, Functions on S to S, Sums, Products andPowers, Peano axioms and Finite induction.2. BINARY RELATIONS :Introduction, Relation Matrices, Algebra of relations, Partial orderings, Equivalence relations and Partitions,Modular numbers, Morphisms, Cyclic unary algebras.UNIT - 2 (20 Hours)3. GRAPH THEORY :Introduction - Definition of a Graph, Simple Graph, Konigsberg bridge problem, Utilities problem, Finite andInfinite graphs, Regular graph, Matrix representation of graphs - Adjacency 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; Hamiltoniancircuit, Hamiltonian path, Complete graph, Traveling salesmen problem. Trees and fundamental circuits, cutsets.UNIT - 3 (25 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, Blockstrutures in ALGOL, The ALGOL grammar.UNIT - 4 (15 Hours)6. BOOLEAN ALGEBRAS :Introduction, Order, Boolean polynomials, Block diagrams for gating networks, Connections with logic, Logicalcapabilities of ALGOL, Boolean applications.Prescribed Text Book :“Modern applied Algebra” by Dr. A. Anjaneyulu, Deepti publications, Tenali.Reference Books :1. Modern applied Algebra by Garrett Birkhoff and Thomas C.Bartee, CBS Publishers and Distributors,Delhi.2. Graph Theory with applications to Engineering and Computer Science by Narsingh Deo, Prentice-Hall ofIndia Pvt. Ltd., New Delhi.

III B.A./B.Sc. MathematicsACHARYA NAGARJUNA UNIVERSITYCURRICULUM - B.A / B.ScMATHEMATICS - PAPER - IV (ELECTIVE - 2)MODERN APPLIED ALGEBRAQUESTION BANK FOR PRACTICALSPaper IV (Elective -2) - CurriculumUNIT - 1 (SETS AND FUNCTIONS, BINARY RELATIONS)1. 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?2. 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 surjection but not injection.6. Show that the Peano’s successor function is an injection but not surjection.7. Find the number of functions from a finite set S of n elements to itself. Among thesei) how many are surjections ii) how many are injections?8. Show that the functions f ( x) = x/and gx ( ) = xfor x∈Rare inverses of one another.9. Prove by induction in Pm , + r= m+ s⇒ r= s.10. If f 1, f 2,......, f mare injections then show that fm o fm−1 o ... o f1is an injection.11. Prove by induction that 3n + 2nis divisible by 3 for all n ≥ 1.nnn ( + 1) 12. Prove by induction that ∑ k = wherek = 1 2n is any positive integer.13. Let X = { a, b} and Y = {, c d,}. e Write down the tabular representation for the relation α on X andY defined by the list : aα c, aα d, aα′ e, bα′ c, bα′ d,bαe.14. 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,bαe.15. Give an example of a relation which is neither reflexive nor irreflexive. Also give its graphical representationand relation matrix.16. If ρ is symmetric, prove that ρ ∨ρ 2n∨...∨ρis symmetric.17. Show that a finite poset has a least element iff it has exactly one minimal element.18. If ρ is reflexive and transitive then show that ρ ∧ ρ is an equivalence relation on a set S .19. Let A= ( S, f )be a finite unary algebra with k elements. Define aRb in A to mean that for somenn∈ N, f ( a) = b. Show that R is reflexive and transitive.220. Let A= [ S, f ] be any finite unary algebra with k elements. Define aRb in A to mean that for somenn∈ N, f ( a) = b. Show that A is cyclic iff for some a∈ S, aRb for all b∈ S .

III B.A./B.Sc. Mathematics3Paper IV (Elective -2) - CurriculumUNIT - 2 (GRAPH THEORY)21. Draw the graph GV ( , E)where V = { A, B, C, D, E}, E = { AB, AC, BC, DE}.22. Find the Edge set E of the graph G = ( V, E)given byBACED23. Explain Konigsberg bridge problem and draw its graph.24. Explain three-utilities problem and draw its graph.25. Draw a graph with the adjacency matrix⎡ 0 1 0 1 0 ⎤⎢1 0 1 1 0⎥⎢⎥⎢ 0 1 0 0 0 ⎥⎢⎥⎢ 1 1 0 0 1 ⎥⎣⎢ 0 0 0 1 0 ⎦⎥⎡2 1 3 0 ⎤⎢1 0 1 2⎥⎢ ⎥26. Draw a graph with the adjacency matrix ⎢3 1 0 1 ⎥⎢ ⎥⎣0 2 1 1 ⎦27. Draw a graph with the incidence matrix28. Find the adjacency matrix of the graph given by⎡1 1 1 0 0 0 0⎤⎢0 0 0 1 1 1 0⎥⎢⎥⎢1 1 0 1 1 0 1 ⎥⎢⎥⎣ 0 0 1 0 0 1 1 ⎦v 4e 8e 1v1e 5 e 7e 4v 3e 2 e3 e 6v 229. Find the adjacency matrix of the graph given byDCAB

44117III B.A./B.Sc. MathematicsPaper IV (Elective -2) - Curriculumv 10 v30. Find the incidence matrix of the graph given byve 8e 7e v 3e 91e 4 ee 6v 5 1e 3e 2 v 231. Find the union, intersection and ring sum of the following two graphs.vh v 6a bavg kv 2cv 3delc fv 4Gv 5G 2132. Show that the two graphs are isomorphic.33. Show that the following graphs are not isomorphic to each other.v 1v u21 u 2v 5 v 6u 5 u 6v 8 v 7u 8 u 7v 4v 3u 4u 334. Draw a circuit from the following graph which is of length nine.v 1v 5v 6 v 2v 9v 8v 4v 335. Draw all the circuits of the following graph.v 2e 1e 2e3e 5e 4e 6v 1 e 7 v 3

III B.A./B.Sc. Mathematics48. Minimise the number of states in the following machine.Present Next state Outputstate a 0a 1a 0a 11 9 1 1 12 2 2 1 03 7 5 0 14 2 2 1 05 2 2 1 06 3 9 1 07 6 8 1 08 9 9 1 09 4 6 0 049. Write the ALGOL expressions for the following Mathematical expressions.3 3i) a −b − 3ab( a+b) ii) AB + C D E iii) π ( r−p)50. Write the ALGOL expressions for the following Mathematical expressions.72Paper IV (Elective -2) - Curriculumn⎛ r ⎞iv) P⎜1+⎟ v)⎝ 100 ⎠log x 2 y 2⎛ + ⎞⎜ ⎟⎝ 2xy⎠2/⎛ y ⎞i) sin2 e x y ii) sin 3 ⎜ x + ⎟ iii) e A − sin A2 iv) − b+ b 2− 4 ac⎝ 2 ⎠2a1v) 1+ sin| x|51. 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×DEvaluate the above for A= 2, B= 3, C= 4, D=5.52. Write the mathematical expression for the following ALGOL expressionsi) sqrt( s× ( s− a) × ( s− b) × ( s−c))ii) sqrt (exp(sin A) −cos( A↑5))iii) − b+ sqrt( b↑2− 4× a× c) / 2×a iv) (exp( x) ↑ x+ exp( 1/ x)/(exp( x↑ 2) + sqrt( x))v) sqrt (exp( A) −sin( A))/x↑253. Write down the effect of the following for statements.i) for i:= 1step 1 until 10 do S ; ii) for i:= −4 step 2 until 7 do S ;iii) for x:= 0 step 01 ⋅ until 1 do S ; iv) for x:= 1step −01 ⋅ until −05 ⋅ do S ;v) for x:= 5 step 1 until 4 do S ;54. Write the effect of executing the assignment statements of the following ALGOL block.begin real abc , , ;c: = 5;enda : = 41 ⋅ ;b: = 2× a+7;c: = 3× a−b;55. Write the ALGOL program which generates an array K with Ki [] = i!for i = 12 , ,....., 10.56. Write ALGOL program to compute the mean of 10 observations.57. Write an ALGOL program for finding the area of a triangle, given its three sides.

III B.A./B.Sc. Mathematics8Paper IV (Elective -2) - Curriculum58. Two one-dimensional arrays X and Y each contain 50 elements. Write the ALGOL program to computeLX=50Σj=1X2j(Length of the vector X ), LY =50Σ Yj=12j(Length of the vector Y )INPROD =50Σj=1XYjj(Inner product of X and Y )⎡a a a59. Write ALGOL program to multiply the matrix A =⎢a a a⎢⎣⎢a a a11 12 1321 22 2331 32 3360. Write down the ALGOL block for finding the roots of ax 2 + bx + c = 0 ( a ≠ 0)⎤⎡b1⎤⎥⎥ by a column matrix B =⎢b⎥⎢2⎥⎦⎥⎣⎢b3⎦⎥, .UNIT - 4 (BOOLEAN ALGEBRAS)61. What is two element Boolean algebra.62. 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 theset { ab , } and also a+ b=sup{ a,}.b66. Given the interval [ ab , ] of a Boolean algebra A . Show that the algebraic system [[ ab , ], ∧∨∗ , , , ab . ] isa 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′)1 2 1 269. 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).71. Write the gating network representing the Boolean expression [( x ∨x ) ∧ x′ ] ∨( x ∧ x ) .72. Write the Boolean expression for the gating network.xy1 2 3 1 2z73. 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 .

III B.A./B.Sc. Mathematics75. Show that ( p′ → p) ∧( p→ p′) is an absurdity.9Paper IV (Elective -2) - Curriculum76. 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)x78. Write an ALGOL program to compute F x ⎧⎪17⋅3− ( x+ 1)x < 5( ) = ⎨2 for⎩⎪ 19⋅ 4 /( 1+ x ) x≥5x ranging from 0 to 10 insteps of 01 ⋅ .x79. Write an ALGOL program to compute F x ⎧⎪3125− x if x < 5( ) = ⎨2 for⎩⎪ ( x− 5)/( 1+ x ) if x≥5x ranging from 1 to 10in steps of 01 ⋅ .80. Write the ALGOL program which computes the relation matrix for the relation ρ 2 where ρ is a binaryrelation on a set X = { x1, x2 ,....., x n}.✦ ✽ ✦

III B.A./B.Sc. Mathematics10Paper IV (Elective -2) - CurriculumACHARYA NAGARJUNA UNIVERSITYB.A / B.Sc. DEGREE EXAMINATION, THEORY MODEL PAPER(Examination at the end of third year, for 2010 - 2011 and onwards)MATHEMATICS PAPER - IV (ELECTIVE - 2)MODERN APPLIED ALGEBRATime : 3 Hours Max. Marks : 1001. State Peano axioms.SECTION - A (6 X 6 = 36 Marks)Answer any SIX questions. Each question carries 6 marks2. Show that the relation m< n means mn | (meaning that m is a divisor of n) is a partial ordering of theset of all positive integers.3. Explain utilities problem.4. If a graph (connected or disconnected) has exactly two vertices of odd degree, prove that theremust be a path joining these two vertices.5. Draw the state diagram for the following machine.Present v ξstate 0 1 0 11 1 2 0 02 2 3 0 03 3 4 0 04 4 1 0 16. Write ALGOL expressions for i)− b +b2 − 4ac2a⎛ x⎞ii) sin 3⎜ ⎟ iii) a b+ cq⎝ y⎠4d7. Define Boolean algebra.8. Prove that in any Boolean algebra, a∧ x =0 and a∨ x =1 imply x = a′ .9.(a)SECTION - B (4 X 16 = 64 Marks)Answer ALL questions. Each question carries 16 marksProve that a function is left invertible iff it is one one.n(b) Prove by induction that Σ k2nn ( + 1)( 2n+1) =where n is any positive integer.k = 16OR10.(a) Prove that an equivalence relation on a set S gives rise to a partition on S.(b)11.(a)If ρ and σ are reflexive and symmetric relations on a set S, then show that the following areequivalent.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.

III B.A./B.Sc. Mathematics11Paper IV (Elective -2) - Curriculum(b)Find the ajacency matrix and incidence matrix of the graph given byv 4e 8e 1v1e 5 e 7e 4v 3e 2 e3 e 6v 212.(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.ORv 2e 1e 2e3e 5e 4e 6v 1 e 7 v 313.(a)(b)14.(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 Outputstate a 0a 1a 0a 11 9 1 1 12 2 2 1 03 7 5 0 14 2 2 1 05 2 2 1 06 3 9 1 07 6 8 1 08 9 9 1 09 4 6 0 0ORExplain i) for statement ii) BLOCK structures in ALGOL.Two one-dimensional arrays X and Y each contain 50 elements. Write the ALGOL program tocompute LX=50Σj=1X2j(Length of the vector X), LY =50Σ Yj=12j(Length of the vector Y),INPROD =50Σj=1XYjj(Inner product of X and Y)

III B.A./B.Sc. Mathematics12Paper IV (Elective -2) - Curriculum15.(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 [ ]( x1∨x2) ∧x3 ∨( x1∧ x2).OR16.(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

III B.A./B.Sc. Mathematics13Paper IV (Elective -2) - CurriculumAACHARYA NAGARJUNA UNIVERSITYB.A / B.Sc. DEGREE EXAMINATION, PRACTICAL MODEL PAPER(Practical examination at the end of third year, for 2010 - 2011 and onwards)MATHEMATICS PAPER - IV (ELECTIVE - 2)MODERN APPLIED ALGEBRATime : 3 Hours Max. Marks : 301Answer ALL questions. Each question carries 72marks. 4× 7 12= 30 M1 (a) If f 1, f 2,......, f mare injections then show that fm o fm−1 o ... o f1is an injection.OR(b) If ρ is reflexive and transitive then show that ρ ∧ ρ is an equivalence relation on a set S .2 (a) Draw a graph with the adjacency matrix⎡ 0 1 0 1 0 ⎤⎢1 0 1 1 0⎥⎢⎥⎢ 0 1 0 0 0 ⎥⎢⎥⎢ 1 1 0 0 1 ⎥⎣⎢0 0 0 1 0 ⎦⎥(b)Show that the two graphs are isomorphic.OR3 (a) Give the state diagram of the following transition table.Present Next state Outputstate 0 1 0 1s 1s 3s 20 0s 2s 1s 41 0s 3s 2s 11 1s 4s 4s 31 0OR(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×DEvaluate the above for A= 2, B= 3, C= 4,D=54 (a) Draw the block diagram of ABC∨ ABC ′ ′ ∨ ABC ′ ′ ′.OR(b) Show that ( p′ → p) ∧( p→ p′) is an absurdity.Written exam : 30 MarksFor record : 10 MarksFor viva-voce : 10 MarksTotal marks : 50 Marks