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1 SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 65 MATERIAL NAME : Problem Material MATERIAL CODE : JM08ADM010 (Sca the above QR code for the direct dowload of this material) Name of the Studet: Brach: Uit I (Logic ad Proofs) Defiitios 1) What are the possible truth values for a atomic statemet? ) Defie a coditioal statemet ad draw the truth table for it 3) Express the statemet, The crop will be destroyed if there is a flood i symbolic form 4) Usig truth table, show that P Q P Q 5) Show that P Q P Q is a tautology without usig the truth table 6) Defie Boud ad Free variables ad give example 7) State ay two rules of iferece with explaatio 8) Give the coverse ad the cotrapositive of the implicatio If it is raiig, the I get wet 9) Express the statemet, Some people who trust others are rewarded i symbolic form 10) What is meat by proof by cotradictio? Predicate Calculus 1) Usig truth table, show that P Q R P Q R P Q R ) Without usig truth tables, show that Q P Q P Q is tautology 3) Without costructig the truth table show that P Q P Q R P Q P R is a tautology 4) Without usig truth tables, show that 5) Show the implicatio without costructig the truth table, Q P P R P P R Q Iferece Theory 1) Show that S is a valid iferece from the premises P Q, Q R, S P ad R P Q R Q R P R R Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 1

2 ) Show that R Sis a valid coclusio from the premises C D, C D H, H A B A B R S ad 3) Prove that the followig premises are icosistet P Q, Q S, R S ad P R 4) Show that the followig premises are icosistet: (i) If Jack misses may classes due to illess, the he fails i high school (ii) If Jack fails i high school, the he is ueducated (iii) If Jack reads a lot of books, the he is ot ueducated (iv) Jack misses may classes due to illess ad reads a lot of books Hit: The give set of premises P1, P, P are said to be cosistet if ad oly if P1 P P T ad the premises are said to be icosistet if ad oly if P1 P P F 5) Show that the followig statemet costitute a valid argumet, by usig method of derivatio If A works hard, the either B or C will ejoy themselves If B ejoys himself, the A will ot work hard If D ejoys himself, the C; will ot Therefore, If A works hard D will ot ejoy himself 6) Symbolize the followig statemets ad the use the method of derivatio If there are meetig, the travellig was difficult If they arrived o time, the travellig was ot difficult They arrived o time Therefore, there was o meetig Show that these statemets costitute a valid argumet 7) Show that the followig premises are icosistet (i) If Vijay misses may classes, the he fails i ME (ii) If Vijay fails i ME, the he is uemployed (iii) If Vijay appears for lot of iterviews, the he is ot uemployed (iv) Vijay misses may classes ad appears for lot of iterviews 8) Verify the validity of the iferece If oe perso is more successful tha the other, the he has worked harder to deserve success Joh has ot worked harder tha Peter Therefore, Joh is ot more successful tha Peter Quatifiers 1) Show that x P( x) Q( x) xq( x) R( x) xp( x) R( x) ) Is the followig coclusio validly derivable from the premises give? If x P( x) Q( x), y P( y) the z Q( z) x P( x) Q( x) x P( x) x Q( x) by idirect method of proof 3) Show that Prepared by CGaesa, MSc, MPhil, (Ph: ) Page

3 Uit II (Combiatorics) Defiitios 1) State the priciple of mathematical iductio ad strog iductio ) Prove by iductio that 3) State ad prove the pigeohole priciple 4) State exteded pigeohole priciple 5) Defie recurrece relatio 3 4 6) Write the geeratig fuctio for the sequece1, a, a, a, a, 7) Fid the recurrece relatio satisfyig the equatio y A(3) B( 4) Small Problems 1) If 13 people are assembled i a room, show that atleast of them must have their birthday i the same day ) Show that if 30 dictioaries i a library cotai a total of 6137 pages, the oe of the dictioaries must have atleast 045 pages 3) Show that C(, r) C( 1, r 1) C( 1, r) Mathematical Iductio ad Strog Iductio ( 1) 1) Use mathematical iductio to prove that 1 3 ( 1)( 1) ) Use mathematical iductio to prove that ( 1) 3) Usig mathematical iductio, prove that 1 3 4) Usig mathematical iductio, prove that ( 1)( 1) ( 1) 3 5) Prove by mathematical iductio, that ( 1) 1 6) Prove that a b is a multiple of a b by usig method of iductio 7) Prove that 8 3 is a multiple of 5 by usig method of iductio 8) Usig mathematical iductio, prove that 3 4 egative itegers 9) Use mathematical iductios to show that ) Use mathematical iductio to prove that 3 7 is divisible by 13, for all o is a multiple of 3, for 1 is divisible by 8, for all 1 Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 3

4 11) State the priciple of strog iductio ad prove that for ay positive iteger is either a prime or a product of primes Recurrece relatios 1) Solve the recurrece relatio u 3 6u 11u 1 6u 0, with u, u 0, u 0 1 ) Solve the recurrece relatio a a a 1, for with a0 1, a1 3) Solve the recurrece relatio a a 0, where 0 ad a0 1, a1 3 4) Usig geeratig fuctio, solve the recurrece relatio a 8a1 15a 0 give that a0, a1 8 5) Use the method of geeratig fuctio to solve the recurrece relatio: S( 1) 8 S( ) 16 S( 1) 4, 1, with S(0) 1 ad S(1) 8 6) Solve S( ) S( 1) 3 S( ) 0, with S(0) 3 ad S(1) 1 by usig geeratig fuctio 7) Solve Y ( ) 7 Y ( 1) 10 Y ( ) 6 8 with Y (0) 1, Y (1) 8) Solve the recurrece relatio a a a for, where a0 3, a1 7 9) Solve, by usig geeratig fuctio, the recurrece relatio y 1 4 y with y for ) Write the recurrece relatio for Fiboacci umber ad hece solve it 11) Fid the geeratig fuctio of Fiboacci sequece F ( ) F( 1) F( ) for with for F(0) F(1) 1 Iclusio ad Exclusio 1) Determie the umber of itegers betwee 1 ad 50 that are divisible by ay of the itegers,3,5 ad 7 by priciple of iclusio exclusio Hit: Let A,B,C ad D be the set of itegers that are divisible by,3,5 ad 7 respectively We have to fid A B C D ) Determie the umber of positive itegers where ad is ot divisible by,3,5 Hit: Let A,B ad C be the set of itegers that are divisible by,3 ad 5 respectively First we have to calculate A B C ad the fid 100 A B C 3) Determie the umber of positive itegers where ad is ot divisible by,3,5 but divisible by 7 Hit: Let A,B,C ad D be the set of itegers that are divisible by,3,5 ad 7 respectively We have to fid AB C D Usig the formula D A B C A B C D A B C D D D A B C ad apply Distributive property Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 4

5 4) Amog the ve itegers, determie the itegers that are divisible by 5 but ot by 7 ad 9 Hit: Let A,B ad C be the set of itegers that are divisible by 5,7 ad 9 respectively We have to fid AB C 5) By usig the priciple of iclusio ad exclusio, fid the umber of primes betwee 41 ad 100? Hit: Let A,B,C ad D be the set of itegers betwee 41 ad 100 that are divisible by,3,5 ad 7 respectively We have to fid A B C D 6) Use the priciple of iclusio ad exclusio to fid the umber of iteger solutios of the system x1 x x3 x4 0, 1 x1 7, 1 x 6, 5 x3 8, x4 9 7) Amog 100 studets, 3 study mathematics, 0 study physics, 45 study biology, 15 study mathematics ad biology, 7 study mathematics ad physics, 10 study physics ad biology ad 30 do ot study ay of the three subjects Fid the umber of studets studyig exactly oe of the three subjects 8) I a survey about likig colours, it was foud that everyoe who was surveyed had a likig for at least oe of the three colours amely R, G ad B Further 30% liked Red, 40% Gree ad 50% Blue Further 10% liked R &G, 5% liked G & B, 10% liked R & B Fid the percetage of surveyed people who liked all the three colours Permutatios ad Combiatios 1) How may differet rearragemets are there of the word REARRANGEMENT? ) How may solutios are there for the equatio x y z 15, where x, y, z 0 3) How may iteger solutios are there x y z 0, subjects to the costraits x 1, y 0, z 4 Hit: The umber of iteger solutios of the equatio x y z d, where x, y, z 0, is C( d o of ukows 1, d) C( d 3 1, d ) (This is Cr formula) Uit III (Graph Theory) Defiitios 1) Defie (a) Graph (b) Simple graph (c) Regular graph ) Defie a complete graph ad give ad example 3) Defie a pedet vertex ad pedet edge 4) Defie matrix represetatio of a graph 5) Defie bipartite graph 6) Defie isomorphism of a graph 7) Defie complemetary ad self complemetary graph 8) Defie coected graph ad give a example for coected ad discoected graphs 9) Fid the umber of coected simple graph with four vertices 10) How may edges are there i a graph with 10 vertices each of degree 5? 11) Defie vertex coectivity 1) Defie edge coectivity Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 5

6 13) Defie Euleria graph 14) Defie Hamiltoia graph 15) Defie a Hamiltoia path of G Icidece ad Adjacecy matrix Isomorphic verificatios Theorems (Refer class ote) (Refer class ote) 1) Discuss the Koigsberg bridge problem ) Prove that i ay graph G, the umber of vertices of odd degree is eve 3) Prove that ay self complemetary graphs have 4 or 4+1 vertices 4) (Whitey s iequality )For ay graph G, show that vertex coectivity edge coectivity miimum vertex degree ad give example for above k 5) If G is k edge coected graph the prove that 6) Prove that a coected graph G is Euleria if ad oly if all the vertices of G are of eve degree 7) Prove that if a graph G has atmost two vertices of odd degree, the there ca be Euler path i G 8) If G is a coected graph with vertices 3 ad if the degree of each vertex is atleast /, the show that G is Hamiltoia Uit IV (Algebraic Structures) Defiitios 1) Defie sub semi groups with example ) Give a example of semi group but ot a mooid 3) Defie semi group homomorphism 4) Defie coset ad give example 5) Defie ormal subgroup of a group 6) Defie kerel of a homomorphism 7) Defie rig ad give example (or) Defie whe a algebraic system S,, rig 8) Defie Itegral domai with example 9) Give a example of a commutative rig without idetity 10) Defie field with example 11) Give a examples of a rig which is ot a field is called a Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 6

7 Small Problems 1) Let E,4,6,8 Show that, ) Prove that the idetity elemet of a group G is uique 3) Prove that the iverse elemet of a group G is uique 4) State ad prove the cacellatio law i a group 5) If a G 1, G be a group, the prove that 1 a * b b * a a, b G 6) I a group G, E ad E, are semi groups but ot mooids a a 7) Fid the multiplicatio iverse of each elemet i Z 11 8) If every elemet i a group is its ow iverse the the group must be abelia 9) For ay a G, a e the prove that G is a abelia 10) Prove that a group ca ot have ay elemet which is idempotet except the idetity elemet a * b a * b a, b G 11) Prove that a group G is abelia if ad oly if 1) Fid all cosets of a sub groups H 1, a of a group G 1, a, a, a 3 4 multiplicatio, where a 1 13) Let G 1, 1, i, i be a group ad 1, 1 uder usual H be a sub group of G What is the umber of distict cosets of H i G 14) Prove that a subgroup of a abelia group is a ormal subgroup Theorems 1) Prove that if G is a abelia group the for all a, b G ad all iteger, a * b b * a ) If S = N X N the set of ordered pairs of positive itegers with the operatio * defied by a, b* c, d ad bc, bd ad if f : S,* Q, is defied by f a, b a / b, the show that f is a semi group homomorphism 3) If S is the set of all ordered pairs ab, of real umbers wit the biary operatio defied by a, b c, d a c, b d, where a, b, c, d are real, prove that S, is a commutative group 4) Show that the set of all positive ratioal umbers forms a abelia group uder the ab compositio defied by a* b 5) O the set Q of all ratioal umbers, the operatio * is defied by a * b a b ab Show that, uder the operatio *, Q is a commutative mooid Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 7

8 6) Prove the ecessary ad sufficiet coditio for a o empty sub set to be a sub group of a group (Or) Prove that a o empty subset H of a group (G,*) is a subgroup if ad oly if ay 1 a, b H implies a * b H 7) Prove that the itersectio of two ormal subgroups of a groups G is also a ormal subgroup of G 8) State ad prove Lagrage s theorem (or) Prove that the order of a subgroup of a fiite group divides the order of the group 9) Prove that the order of ay elemet of a fiite group is a divisor of the order of the group (ie) Oa ( ) is a divisor of OG ( ) for all a G OG ( ) 10) If G is a fiite group, the prove that a efor ay elemet a G 11) Prove that a subgroup H of a group G uder the operatio * is a ormal subgroup if ad oly if a 1 * h* a H for every a G ad h H 1) Prove that a subgroup H of a group G is ormal if ad oly if xhx 1 H for all x G (or) Prove that A subgroup H of G is ormal if ad oly if left coset of H i G is equal to the right coset of H i G 13) If N ad M are ormal subgroup of G, prove that NM is also a ormal subgroup of G 14) Let,* G ad, G be groups ad f is homomorphism from G to G, the prove that the kerel of f is a ormal subgroup 15) If f is a homomorphism of G oto G with kerel K, the G/ K is isomorphic to G 16) Prove that every fiite group of order is isomorphic to a permutatio group of degree (Cayley s theorem o permutatio group) Uit V (Lattices ad Boolea algebra) Defiitios 1) Defie a poset ad give a example (or) Defie partially ordered set ) Defie a distributio lattice 3) Give a example of a lattice which is modular but ot distributive 4) State ay two properties of Lattices 5) Defie Boolea Algebra 6) I a Boolea Algebra, prove that the complemet of ay elemet is uique 7) Show that i ay Boolea algebra, ( a b)( a c) ac ab bc 8) Show that absorptio laws are valid i a Boolea algebra 9) Give a example of two elemet Boolea Algebra Lattices Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 8

9 1) Let N be the set of all atural umbers with the relatio R as follows: x R y if ad oly if x divides y Show that R is a partial order relatio o N ) Let N be set of all atural umbers ad defie m if mis a o egatie iteger Show that N, is a poset 3) I a Lattice L,, prove that X Y Z X Y X Z 4) If L, is a Lattice, the for ay,, a b c a b a c a b c L prove that 5) Show that i a complemeted distributive lattice, the De Morga s laws hold (or) If L,, be a complemeted distributive lattice, the for ay a, b L, prove (1) a b a b () a b a b 6) Show that i a complemeted, distributed lattice, a b a b 0 a b 1 b a 7) If L, be a lattice, prove the followig equivalet a, b L, a b a b a a b b 8) If a, b, c are elemet of a distributive lattice L,, show that a b a c ad a b a c b c 9) If L, is a ordered lattice, show that L,, is a algebraic lattice 10) Show that every chai is a distributive lattices 11) Show that if L is a distributive lattice tha for all a, b, c L, a* b b* c c* a a b* b c* c a 1) If L is a distributive lattice with 0 ad 1, show that each elemet has at most oe complemet 13) Show that every distributive lattice is modular Is the coverse true? Justify the claim Boolea Algebra 1) Show that i ay Boolea algebraa ba c ac ab bc ) Show that i ay Boolea algebra, a bif ad oly if abab 0 3) I a Boolea algebra B prove that a b a b ad all a, b B (DeMorgo s Law ) a b a bfor 4) I ay Boolea algebra, show thata bb cc a a bb cc a 5) Prove that i ay Boolea algebra ab bc ca ab bc ca 6) If x, yare elemets i a Boolea algebra, the prove that x y x y 7) If B is a Boolea algebra, the for a B a 1 1, a0 0 Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 9

10 ---- All the Best ---- Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 10