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1 SUBJECT NAME : Discrete Mthemtics SUBJECT CODE : MA 2265 MATERIAL NAME : Formul Mteril MATERIAL CODE : JM08ADM009 Nme of the Student: Brnch: Unit I (Logic nd Proofs) 1) Truth Tble: Conjunction Disjunction Conditionl Biconditionl q q q q q q q q T T T T T T T T T T T T T F F T F T T F F T F F F T F F T T F T T F T T F F F F F F F F T F F T Negtion T F F T 2) Tutology nd Contrdiction: A Comound roosition P ( P P P ) = 1, 2,... n where 1 2 tutology if it is true for every truth ssignment for P1, P2,... P n. P, P,... Pn vribles re clled P is clled Contrdiction if it is flse for every truth ssignment for P1, P2,... P n. If roosition is neither tutology nor Contrdiction is clled contingency. 3) Lws of lgebr of roosition: Nme of Lw Priml form Dul form Idemotent lw Identity lw F T Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 1

2 Dominnt lw T T F F Comlement lw Commuttive lw T F q q q q Associtive lw ( q) r ( q r ) ( q) r ( q r ) Distributive lw ( q r ) ( q) ( r ) ( q r ) ( q) ( r ) Absortion lw ( q) ( ) Demorgn s lw ( q) q ( ) q q q Double Negtion lw 4) Equivlence involving Conditionls: Sl.No. Proositions 1. q q 2. q q 3. q q 4. ( q) ( r ) ( q r ) r q r q r 5. ( ) ( ) ( ) 5) Equivlence involving Biconditionls: Sl.No. Proositions 1. q ( q) ( q ) 2. q q 3. q ( q) ( q) Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 2

3 4. ( ) q q 6) Tutologicl Imliction: A B if nd only if A B is tutology. (i.e) To rove A B, it enough to rove A B is tutology. 7) The Theory of Inferences: The nlysis of the vlidity of the formul from the given set of remises by using derivtion is clled theory of inferences 8) Rules for inferences theory: Rule P: A given remise my be introduced t ny stge in the derivtion. Rule T: A formul S my be introduced in derivtion if S is tutologiclly imlied by one or more of the receding formule in the derivtion. Rule CP: If we cn drive S from R nd set of given remises, then we cn derive R S from the set of remises lone. In such cse R is tken s n dditionl remise (ssumed remise). Rule CP is lso clled the deduction theorem. 9) Indirect Method of Derivtion: Whenever the ssumed remise is used in the derivtion, then the method of derivtion is clled indirect method of derivtion. 10) Tble of Logicl Imlictions: Nme of Lw Priml form Simlifiction Addition Disjunctive Syllogism q q q q q q ( ) q q ( ) q q q q Modus Ponens ( ) Modus Tollens ( ) q q q q r r Hyotheticl Syllogism ( ) ( ) Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 3

4 q q Unit II (Combintorics) 1) Princile of Mthemticl Induction: Let P( n) be sttement or roosition involving for ll ositive integers n. Ste 1: P (1) is true. Ste2: Assume tht P( k ) is true. Ste3: We hve to rove P( k + 1) is true. 2) Princile of Strong induction. Let P( n ) be sttement or roosition involving for ll ositive integers n. Ste 1: P (1) is true. Ste2: Assume tht P( n ) is true for ll integers 1 n k. Ste3: We hve to rove P( k + 1) is true. 3) The Pigeonhole Princile: If n igeons re ssigned to m igeonholes nd m < n, thn t lest one igeonhole contins two or more igeons. 4) The Extended Pigeonhole Princile: If n igeons re ssigned to m igeonholes thn one igeonhole must contins t lest ( n 1) + 1igeons. m 5) Recurrence reltion: An eqution tht exresses n, the generl term of the sequence{ n } in terms of one or more of the revious terms of the sequence, nmely 0, 1,... n 1, for ll integers n is clled recurrence reltion for{ n } or difference eqution. 6) Working rule for solving homogeneous recurrence reltion: Ste 1: The given recurrence reltion of the form C ( n) + C ( n) C ( n) = 0 0 n 1 n 1 k n k Ste 2: Write the chrcteristic eqution of the recurrence reltion C0rn + k + C1r ( 1 ) Ckrn = 0 n + k + Ste 3: Find ll the roots of the chrcteristic eqution nmely r 1, r 2,... r k. Ste 4: n Cse (i): If ll the roots re distinct then the generl solution is = b r + b r + + b r n n n n k k Cse (ii): If ll the roots re equl then the generl solution is 2 ( ) = b + nb + n b + r n Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 4

5 Unit III (Grh Theory) 1) Grh: A grh G=(V,E) consists of two sets V { v1, v2,... v υ } = {,,... }, clled the set of edges of G. E e e e 1 2 =, clled the set of vertices nd 2) Simle grh: A grh is sid to be simle grh if it hs no loos nd rllel edges. Otherwise it is multi grh. 3) Regulr grh: If every vertex of simle grh hs the sme degree, then the grh is clled regulr grh. If every vertex in regulr grh hs degree n, then the grh is clled n-regulr. 4) Comlete grh: A simle grh in which ech ir of distinct vertices is joined by n edge is clled comlete grh. The comlete grh on n vertices is denoted by K n. 5) Pendent vertex nd Pendent edge: A vertex with degree one is clled endent vertex nd the only edge which is incident with endent vertex is clled the endent edge. 6) Mtrix reresenttion of grh: There re two wys of reresenting grh by mtrix nmely djcent mtrix nd incidence mtrix s follows: Adjcency mtrices: Let G be grh with n vertices, then the djcency mtrix, AG ( Aij ) = defined 1, if ui, v j re djcent by Aij =. 0, otherwise Incidence mtrix: Let G be grh with n vertices, then the incidence mtrix of G is n n x e mtrix B G ( Bij ) = defined by B ij th th 1, if j edge is incident on the i vertex = 0, otherwise 7) Birtite grh: A grh G=(V,E) is clled birtite grh if its vertex set V cn be rtitioned into two subsets V1nd V2such tht ech edge of G connects nd vertex of V 1 to vertex of V 2. In other words, no edge joining two vertices, in V 1 or two vertices in V 2. 8) Isomorhism of grh: The simle grhs G = ( V, E ) nd G ( V, E ) = re isomorhic if there is one to one nd onto function f from V1to V 2 with the roerty tht nd b re djcent in G1if nd only if f ( ) nd f ( b ) re djcent in G 2, for ll nd b in V 1. Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 5

6 9) Comlementry nd Self comlementry grh: Let G be grh. The comlement G of G is defined by ny two vertices re djcent in G if nd only if they re not djcent ing. G is sid to be self comlementry grh if G is isomorhic tog. 10) Connected grh: A grh G is sid to be connected if there is t lest one th between every ir of vertices in G. Otherwise G is disconnected. A disconnected grh consists of two or more connected sub grhs nd ech of them is clled comonent. It is denoted by ω ( G). 11) Cut edge: A cut edge of grh G is n edge e such tht ω ( G e) > ω ( G).(i.e) If G is connected nd e is cut edge of G, then G eis disconnected. 12) Cut vertex: A cut vertex of grh G is vertex v such ω ( G v) > ω ( G).(i.e) If G is connected nd v is cut vertex of G, then G vis disconnected. 13) Define vertex connectivity. The connectivity κ ( G) of G is the minimum k for which G hs k-vertex cut. If G is either trivil or disconnected then κ ( G) = 0. 14) Define edge connectivity. The edge connectivity κ ( G) of G is the minimum k for which G hs k-edge cut. If G is either trivil or disconnected then κ ( G) = 0 15) Define Eulerin grh. A th of grh G is clled n Eulerin th, if it includes ech edge of G exctly once. A circuit of grh G is clled n Eulerin circuit, if it includes ech edge of G exctly one. A grh contining n Eulerin circuit is clled n Eulerin grh. 16) Define Hmiltonin grh. A simle th in grh G tht sses through every vertex exctly once is clled Hmilton th. A circuit in grh G tht sses through every vertex exctly once is clled Hmilton circuit. A grh contining Hmiltonin circuit. Unit IV (Algebric Structures) 1) Semi grou: If G is non-emty set nd * be binry oertion on G, then the lgebric system ( G,*) is clled semi grou, if G is closed under * nd * is ssocitive. Exmle: If Z is the set of ositive even numbers, then ( Z, + ) nd (, ) grous. Z re semi Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 6

7 2) Monoid: If semi grou ( G,*) hs n identity element with resect to the oertion *, then ( G,*) is clled monoid. It is denoted by ( G,*, e ). Exmle: If N is the set of nturl numbers, then ( N, + ) nd (, ) the identity elements 0 nd 1 resectively. ( Z, + ) nd (, ) out monoids, where Z is the set of ll ositive even numbers 3) Sub semi grous: If ( G,*) is semi grou nd H G clled sub semi grou of ( G,*). N re monoids with Z re semi grous with is clled under the oertion *, then ( H,*) Exmle: If the set Eof ll even non-negtive integers, the ( E, + ) is sub semi grou of the semi grou ( N, + ), where N is the set of nturl numbers. 4) Semi grou homomorhism: If ( G,* ) nd (, ) G re two semi grous, then ming f : G G is clled semi grou homomorhism, if for ny, b G, f ( * b) = f ( ) f ( b). A homomorhism f is clled isomorhism if f is 1-1 nd onto. 5) Grou: If G is non-emty set nd * is binry oertion of G, then the lgebric system ( G,*) is clled grou if the following conditions re stisfied. (i) (ii) (iii) (iv) Exmle: (, ) 6) Abelin grou: Closure roerty Associtive roerty Existence of identity element Existence of inverse element Z + is grou nd (, ) Zi is not grou. A grou ( G,*), in which the binry oertion * is commuttive, is clled commuttive grou or belin grou. Exmle: The set of rtionl numbers excluding zero is n belin grou under the multiliction. 7) Coset: If H is subgrou of grou G under the oertion *, then the set H, where G, define by H = { * h / h H} = is clled the left coset of H in G generted by the element G. Similrly the set H is clled the right coset of H in G generted by the element G. is Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 7

8 Exmle: G = { 1, 1, i, i} be grou under multiliction nd H = { 1, 1} is subgrou of G. The right cosets re 1H = { 1, 1}, 1H = { 1,1}, ih = { i, i} { i, i} ih =. 8) Lgrnge s theorem: The order of ech subgrous of finite grou is divisor of order of grou. 9) Cyclic grou: = nd A grou ( G,*) is sid to be cyclic, if nd element G such tht every element of 2 n G generted by. (i.e) G =< >= { 1,,,... = e} Exmle: G = { 1, 1, i, i} =< >= =. = is cyclic grou under the multiliction. The genertor is i, becuse i = 1, i = 1, i, i = i. 10) Norml subgrou: A subgrou H of the grou G is sid to be norml subgrou under the oertion *, if for ny G, H = H. 11) Kernel of homomorhism: If f is grou homomorhism from ( G,*) nd (, ) G, then the set of element of G, which re med into e, the identity element of G, is clled the kernel of the homomorhism f nd denoted by ker ( f ). 12) Fundmentl theorem of homomorhism: If f is homomorhism of G on to G with kernel K, then G / K is isomorhic to G. 13) Cyley s theorem: Every finite grou of order n is isomorhic to ermuttion grou of degree n. 14) Ring: An lgebric system ( S, +i, ) is clled ring if the binry oertions +nd i on S stisfy the following roerties. (i) ( S, + ) is n belin grou (ii) ( Si, ) is semi grou (iii) The oertion i is distributive over +. Exmle: The set of ll integers Z, nd the set of ll rtionl numbers R re rings under the usul ddition nd usul multiliction. 15) Integrl domin: A commuttive ring without zero divisor is clled Integrl domin. Exmle: (i) ( R, +i, ) is n integrl domin, since, 0, b 0then b 0. (ii) ( Z, +, ) b R such tht + is not n integrl domin, becuse 2, 3 Z10 nd = 0. Therefore 2 nd 5 re zero divisors. Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 8

9 16) Field: A commuttive reing ( S, +i, ) which hs more thn one element such tht every nonzero element of S hs multilictive inverse in S is clled field. Exmle: The ring of rtionl numbers ( Q, +i, ) is field since it is commuttive ring nd ech non-zero element is inversible. Unit V (Lttices nd Boolen lgebr) 1) Prtilly ordered set (Poset): A reltion R on set A is clled rtil order reltion, if R is reflexive, ntisymmetric nd trnsitive. The set A together with rtil order reltion R is clled rtilly ordered set or oset. Exmle: The greter thn or equl to ( ) reltion is rtil ordering on the set of integers Z. 2) Lttice: A lttice is rtilly ordered set ( L, ) in which every ir of elements, glb nd lub. 3) Generl formul: glb, b = * b = b i) { } ii) l b {, } iii) iv) 4) Proerties: u b = b = b * b & * b b b * b = If b = b & b b b Nme of Lw Priml form Dul form Idemotent lw * = = b Lhs Commuttive lw * b = b* b = b Associtive lw ( * b) * c = *( b* c) ( b) c = ( b c) Distributive lw * ( q r ) ( * q) ( * r ) ( q * r ) ( q) *( r ) Absortion lw *( b) = ( * ) b = Comlement * = 0 = 1 Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 9

10 Demorgn s lw ( * ) b = b b = b = ( ) = * Double Negtion lw 5) Comlemented Lttices: A Lttice ( L,*, ) is sid to be comlemented if for ny L, there exist L, such tht * = 0nd = 1. 6) Demorgn s lws: Let ( L,*, ) be the comlemented lttice, then ( * ) b = b. ( ) = * 7) Comlete Lttice: A lttice ( L,*, ) lub. 8) Lttice Homomorhism: Let (,*, ) b = b nd is comlete if for ll non-emty subsets of L, there exists glb nd L nd ( S,, ) be two lttices. A ming g : L Sis clled lttices homomorhism if g( * b) = g( ) g( b) nd g( b) = g( ) g( b). 9) Modulr Lttice: b c L A lttice ( L,*, ) is sid to be modulr if for ny,, i) c ( b* c) = ( b) * c ii) c *( b c) = ( * b) c 10) Chin in Lttice: Let ( L, ) be Chin if i) b or c nd ii) b nd c 11) Condition for the lgebric lttice: A lttice ( L,*, ) is sid to be lgebric if it stisfies Commuttive Lw, Associtive Lw, Absortion Lw nd Existence of Idemotent element. 12) Isotone roerty: Let ( L,*, ) roerty if be lttice. The binry oertions *nd re sid to ossess isotone b c * b * c b c. Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 10

11 13) Boolen Algebr: A Boolen lgebr is lttice which is both comlemented nd distributive. It is denoted by( B,*, ) All the Best ---- Prered by C.Gnesn, M.Sc., M.Phil., (Ph: ) Pge 11