SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 65 MATERIAL NAME : Part A questios MATERIAL CODE : JM08AM1013 REGULATION : R008 UPDATED ON : May-Jue 016 (Sca the above Q.R code for the direct dowload of this material) Name of the Studet: Brach: Uit I (Logic ad Proofs) 1. Defie Tautology with a example.. Defie a rule of Uiversal specificatio. 3. Fid the truth table for the statemet P Q. 4. Costruct a truth table for the compoud propositio ( p q) ( q p) 5. Usig truth table show that P ( P Q) P.. 6. Costruct the truth table for the compoud propositio ( p q) ( p q) 7. Give P = {, 3,4,5,6}, state the truth value of the statemet ( x P )( x + 3 = 10) + =.. 8. Show that the propositios p qad p q are logically equivalet. 9. Show that ( p r ) ( q r ) ad ( ) 10. Prove that p, p q, q r r. p q r are logically equivalet. 11. Without usig truth table show that P ( Q P ) P ( P Q) ( P Q R ) (( P Q) ( P R) ) 1. Show that ( ). is a tautology. 13. Usig truth table, show that the propositio p ( p q) is a tautology. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph:9841168917) Page 1
( ) p p q q a tautology? 14. Is ( ) 15. Write the egatio of the statemet( x)( y) p( x, y). 16. What are the egatios of the statemets x ( x > x) ad x ( x = ) =? 17. Give a idirect proof of the theorem If 3 + is odd, the is odd. 18. Whe do you say that two compoud propositios are equivalet? 19. What are the cotra positive, the coverse ad the iverse of the coditioal statemet If you work hard the you will be rewarded. 0. Symbolically express the followig statemet. It is ot true that 5 star ratig always meas good food ad good service. Uit II (Combiatorics) 1. State the priciple of strog iductio.. Use mathematical iductio to show that + 1!, = 1,, 3,... =. ( + 1) 3. Use mathematical iductio to show that 1 + + 3 +... + =. 4. If seve colours are used to pait 50 bicycles, the show that at least 8 bicycles will be the same colour. 5. Write the geeratig fuctio for the sequece 6. Fid the recurrece relatio for the Fiboacci sequece. 3 4 1, a, a, a, a,... 7. Fid the recurrece relatio satisfyig the equatio y = A(3) + B( 4). 8. Solve the recurrece relatio y ( k ) y ( k ) y ( k ) y () = 16ad y (3) = 80. 9. Solve: ak = 3ak 1, for k 1, with a 0 =. 8 1 + 16 = 0for k, where 10. Fid the umber of o-egative iteger solutios of the equatio x 1 + x + x 3 = 11. 11. State Pigeohole priciple. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph:9841168917) Page
1. What is well orderig priciple? 13. I how may ways ca all the letters i MATHEMATICAL be arraged. 14. What is the umber of arragemets of all the six letters i the word PEPPER? 15. How may permutatios of { a, b, c, d, e, f, g } ad with a? 16. How may differet bit strigs are there of legth seve? C, = C, +. 17. Show that ( ) ( ) Uit III (Graph Theory) 1. Defie Pseudo graph.. Defie complete graph ad give a example. 3. Defie a regular graph. Ca a complete graph be a regular graph? 4. Whe is a simple graph G bipartite? Give a example. 5. Defie a coected graph ad a discoected graph with examples. 6. Defie strogly coected graph. 7. Is the directed graph give below strogly coected? Why or why ot? 8. Defie complete bipartite graph. 9. Draw a complete bipartite graph of K,3ad K 3,3. 10. Defie isomorphism of two graphs. 11. Defie self complemetary graph. 1. Give a example of a Euler graph. 13. Give a example of a o-euleria graph which is Hamiltoia. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph:9841168917) Page 3
14. Give a example of a graph which is Euleria but ot Hamiltoia. 15. State the hadshakig theorem. 16. State the ecessary ad sufficiet coditios for the existece of a Euleria path i a coected graph. 17. Defie complemetary graph G of a simple graph G. If the degree sequece of the simple graph is 4, 3, 3,,, what is the degree sequece of G. 18. For which value of m ad does the complete bipartite graph K m, have a (i) Euler circuit (tour) (ii) Hamilto circuit (cycle). 19. Draw the graph represeted by the give adjacecy matrix 0 1 0 1 1 0 1 0. 0 1 0 1 1 0 1 0 0. Obtai the adjacecy matrix of the graph give below. Uit IV (Algebraic Structures) 1. Defie a semigroup.. Defie mooids. 3. State ay two properties of a group. 4. Prove that idetity elemet is uique i a group. 5. Whe is a group( G,*) called abelia? 6. Prove that every subgroup of a abelia group is ormal. 7. Prove that the idetity of a subgroup is the same as that of the group. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph:9841168917) Page 4
8. Defie homomorphism ad isomorphism betwee two algebraic systems. 9. Give a example for homomorphism. 10. If aad bare ay two elemets of a group G,*, show that Gis a abelia group if a * b = a * b. ad oly if ( ) 11. State Lagrage s theorem i group theory. 1. Show that every cyclic group is abelia. 13. Let M,*, e M be a mooidad a M. If aivertible, the show that its iverse is uique. 14. If ' a' is a geerator of a cyclic group G, the show that 15. Show that the set of all elemets aof a group (, ) x Gis a subgroup of G. 1 a is also a geerator of G. G such that a x = x afor every 16. Obtai all the distict left cosets of { [ 0 ],[ 3 ] } i the group ( 6, 6 ) uio. 17. Defie rig ad give a example. 18. Defie a field i a algebraic system. 19. Give a example of a rig which is ot a field. 0. Defie a commutative rig. Uit V (Lattices ad Boolea algebra) 1. Draw the Hasse diagram of, relatio be such that x X, where {,4,5,10,1, 0,5} y is xad y. Z + ad fid their X = ad the. Let X = { 1,, 3,4,6,8,1,4} ad R be a divisio relatio. Fid the Hasse diagram of the poset X, R. = ad ρ ( A) be its power set. Draw a Hasse diagram of ρ( A),. 3. Let A { a, b, c} 4. Show that least upper boud of a subset B i a poset ( A, ) is uique is it exists. 5. Defie a lattice. Give suitable example. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph:9841168917) Page 5
6. Defie sub-lattice. 7. Whe a lattice is called complete? 8. Whe is a lattice said to be bouded? 9. Defie lattice homomorphism. 10. Show that i a distributive lattice, if complemet of a elemet exits the it must be uique. 11. Give a example of a distributive lattice but ot complemeted. 1. I a Lattice ( L, ), prove that ( ) 13. Show that i a lattice if a b c, the a b = b c a a b = a, for all a, b L. = ad( a b) ( b c) = ( a b) ( b c) =. 14. Check whether the posets {( ) } Justify your claim. 15. Defie a Boolea algebra. { } 1, 3,6,9, D ad ( ) 16. Whe is a lattice said to be a Boolea algebra? 17. Is there a Boolea algebra with five elemets? Justify your aswer. 18. Prove the Boolea idetity: a. b + a. b = a. 19. Show that i a Boolea algebra ab + a b = 0if ad oly if a = b. 0. Show that the absorptio laws are valid i a Boolea algebra. 1,5, 5,15, D are lattices or ot. ---- All the Best---- Prepared by C.Gaesa, M.Sc., M.Phil., (Ph:9841168917) Page 6