Name of the Student: Unit I (Logic and Proofs) 1) Truth Table: Conjunction Disjunction Conditional Biconditional

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1 SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 2265 MATERIAL NAME : Formula Material MATERIAL CODE : JM08ADM009 (Sca the above QR code for the direct dowload of this material) Name of the Studet: Brach: Uit I (Logic ad Proofs) 1) Truth Table: Cojuctio Disjuctio Coditioal Bicoditioal p q p q p q p q p q p q p q p q T T T T T T T T T T T T F F T F T T F F T F F T F F T T F T T F T F F F F F F F F T F F T F F T Negatio p p T F F T 2) Tautology ad Cotradictio: A Compoud propositio P P P P 1, 2, where P1 P2 tautology if it is true for every truth assigmet for P1, P2, P,, P variables are called P is called a Cotradictio if it is false for every truth assigmet for P1, P2, P If a propositio is either a tautology or a Cotradictio is called cotigecy 3) Laws of algebra of propositio: Name of Law Primal form Dual form Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 1

2 Idempotet law p p p p p p Idetity law p F p Domiat law p T T p T p p F F Complemet law Commutative law p p T p p F p q q p p q q p Associative law p q r p q r p q r p q r Distributive law p q r p q p r p q r p q p r p p q p Absorptio law p p q p Demorga s law p q p q p q q p Double Negatio law p p 4) Equivalece ivolvig Coditioals: SlNo Propositios p q p q p q q p p q p q 4 p q p r p q r 5 p r q r p q r 5) Equivalece ivolvig Bicoditioals: SlNo Propositios p q 1 p q q p 2 p q p q Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 2

3 p q 3 p q p q 4 p q p q 6) Tautological Implicatio: A B if ad oly if A B is tautology (ie) To prove A B, it eough to prove A B is tautology 7) The Theory of Ifereces: The aalysis of the validity of the formula from the give set of premises by usig derivatio is called theory of ifereces 8) Rules for ifereces theory: Rule P: A give premise may be itroduced at ay stage i the derivatio Rule T: A formula S may be itroduced i a derivatio if S is tautologically implied by oe or more of the precedig formulae i the derivatio Rule CP: If we ca drive S from R ad a set of give premises, the we ca derive R S from the set of premises aloe I such a case R is take as a additioal premise (assumed premise) Rule CP is also called the deductio theorem 9) Idirect Method of Derivatio: Wheever the assumed premise is used i the derivatio, the the method of derivatio is called idirect method of derivatio 10) Table of Logical Implicatios: Simplificatio Name of Law Primal form p q p p q q Additio Disjuctive Syllogism p p q q p q p p q q q p q p p Modus Poes p q q Modus Tolles p q q p Hypothetical Syllogism p q q r p r Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 3

4 p q q p Uit II (Combiatorics) 1) Priciple of Mathematical Iductio: Let P ( ) be a statemet or propositio ivolvig for all positive itegers Step 1: P (1) is true Step2: Assume that Pk ( ) is true Step3: We have to prove Pk ( 1) is true 2) Priciple of Strog iductio Let P ( ) be a statemet or propositio ivolvig for all positive itegers Step 1: P (1) is true Step2: Assume that P ( ) is true for all itegers 1 k Step3: We have to prove Pk ( 1) is true 3) The Pigeohole Priciple: If pigeos are assiged to m pigeoholes ad m, tha at least oe pigeohole cotais two or more pigeos 4) The Exteded Pigeohole Priciple: If pigeos are assiged to m pigeoholes tha oe pigeohole must cotais at least 1 1 pigeos m 5) Recurrece relatio: A equatio that expresses a, the geeral term of the sequecea Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 4 i terms of oe or more of the previous terms of the sequece, amely a 0, a 1, a 1, for all itegers is called a recurrece relatio fora or a differece equatio 6) Workig rule for solvig homogeeous recurrece relatio: Step 1: The give recurrece relatio of the form C ( ) a C ( ) a C ( ) a k k Step 2: Write the characteristic equatio of the recurrece relatio C0r k C1r C 0 1 kr k Step 3: Fid all the roots of the characteristic equatio amely r 1, r 2, r k Step 4: Case (i): If all the roots are distict the the geeral solutio is a b r b r b r k k Case (ii): If all the roots are equal the the geeral solutio is a b b b r

5 Uit III (Graph Theory) 1) Graph: A graph G=(V,E) cosists of two sets V v1, v2, v,,, called the set of edges of G E e1 e2 e, called the set of vertices ad 2) Simple graph: A graph is said to be simple graph if it has o loops ad parallel edges Otherwise it is multi graph 3) Regular graph: If every vertex of a simple graph has the same degree, the the graph is called a regular graph If every vertex i a regular graph has degree, the the graph is called -regular 4) Complete graph: A simple graph i which each pair of distict vertices is joied by a edge is called a complete graph The complete graph o vertices is deoted by K 5) Pedet vertex ad Pedet edge: A vertex with degree oe is called a pedet vertex ad the oly edge which is icidet with a pedet vertex is called the pedet edge 6) Matrix represetatio of a graph: There are two ways of represetig a graph by a matrix amely adjacet matrix ad icidece matrix as follows: Adjacecy matrices: by A ij Let G be a graph with vertices, the the adjacecy matrix, AG Aij 1, if ui, vj are adjacet 0, otherwise defied Icidece matrix: Let G be a graph with vertices, the the icidece matrix of G is a x e matrix B G ij B defied by B ij th th 1, if j edge is icidet o the i vert ex 0, otherwise 7) Bipartite graph: A graph G=(V,E) is called a bipartite graph if its vertex set V ca be partitioed ito two subsets V1 ad V2 such that each edge of G coects ad vertex of V 1 to a vertex of V 2 I other words, o edge joiig two vertices, i V 1 or two vertices i V 2 8) Isomorphism of a graph: The simple graphs G V, E ad G V, E are isomorphic if there is a oe to oe ad oto fuctio f from V1 to V 2 with the property that a ad b are adjacet i G1 if ad oly if f( a) ad f( b ) are adjacet i G 2, for all a ad b i V 1 Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 5

6 9) Complemetary ad Self complemetary graph: Let G be a graph The complemet G of G is defied by ay two vertices are adjacet i G if ad oly if they are ot adjacet ig G is said to be a self complemetary graph if G is isomorphic tog 10) Coected graph: A graph G is said to be coected if there is at least oe path betwee every pair of vertices i G Otherwise G is discoected A discoected graph cosists of two or more coected sub graphs ad each of them is called a compoet It is deoted by ( G) 11) Strogly Coected ad Weakly Coected graph: 12) Cut edge: A cut edge of a graph G is a edge e such that ( G e) ( G) (ie) If G is coected ad e is a cut edge of G, the G eis discoected 13) Cut vertex: A cut vertex of a graph G is a vertex v such ( G v) ( G) (ie) If G is coected ad v is a cut vertex of G, the G vis discoected 14) Defie vertex coectivity The coectivity ( G) of G is the miimum k for which G has a k-vertex cut If G is either trivial or discoected the ( G) 0 15) Defie edge coectivity The edge coectivity ( G) of G is the miimum k for which G has a k-edge cut If G is either trivial or discoected the ( G) 0 16) Defie Euleria graph A path of graph G is called a Euleria path, if it icludes each edge of G exactly oce A circuit of a graph G is called a Euleria circuit, if it icludes each edge of G exactly oe A graph cotaiig a Euleria circuit is called a Euleria graph 17) Defie Hamiltoia graph A simple path i a graph G that passes through every vertex exactly oce is called a Hamilto path A circuit i a graph G that passes through every vertex exactly oce is called a Hamilto circuit A graph cotaiig a Hamiltoia circuit Uit IV (Algebraic Structures) 1) Semi group: If G is a o-empty set ad * be a biary operatio o G, the the algebraic system G,* is called a semi group, if G is closed uder * ad * is associative Example: If Z is the set of positive eve umbers, the Z, ad Z, are semi groups Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 6

7 2) Mooid: If a semi group G,* has a idetity elemet with respect to the operatio *, the G,* is called a mooid It is deoted by G,*, e Example: If N is the set of atural umbers, the N, ad N, are mooids with the idetity elemets 0 ad 1 respectively Z, ad Z, are semi groups with out mooids, where Z is the set of all positive eve umbers 3) Sub semi groups: If G,* is a semi group ad H Gis called uder the operatio *, the H,* is called a sub semi group of G,* Example: If the set E of all eve o-egative itegers, the E, is a sub semi group of the semi group N,, where N is the set of atural umbers 4) Semi group homomorphism: If G,* ad G, are two semi groups, the a mappig f : G G is called a semi group homomorphism, if for ay a, b G, f ( a * b) f ( a) f ( b) A homomorphism f is called isomorphism if f is 1-1 ad oto 5) Group: If G is a o-empty set ad * is a biary operatio of G, the the algebraic system G,* is called a group if the followig coditios are satisfied (i) Closure property (ii) Associative property (iii) Existece of idetity elemet (iv) Existece of iverse elemet Example: Z, is a group ad Z, is ot a group 6) Abelia group: A group G,*, i which the biary operatio * is commutative, is called a commutative group or abelia group Example: The set of ratioal umbers excludig zero is a abelia group uder the multiplicatio 7) Coset: If H is a subgroup of a group G uder the operatio *, the the set ah, where a G, defie by ah a* h / h H is called the left coset of H i G geerated by the elemet a G Similarly the set Ha is called the right coset of H i G geerated by the elemet a G Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 7

8 Example: G 1, 1, i, i be a group uder multiplicatio ad H 1, 1 is a subgroup of G The right cosets are 1H 1, 1, 1H 1,1, ih i, i ih i, i ad 8) Lagrage s theorem: The order of each subgroups of a fiite group is a divisor of a order of a group 9) Cyclic group: A group G,* is said to be cyclic, if ad elemet a G such that every elemet of 2 G geerated by a (ie) G a 1, a, a, a e Example: G 1, 1, i, i is a cyclic group uder the multiplicatio The geerator is i, because i 1, i 1, i, i i 10) Normal subgroup: A subgroup H of the group G is said to be ormal subgroup uder the operatio *, if for ay a G, ah Ha 11) Kerel of a homomorphism: If f is a group homomorphism from,* G ad, G, the the set of elemet of G, which are mapped ito e, the idetity elemet of G, is called the kerel of the homomorphism f ad deoted by ker f 12) Fudametal theorem of homomorphism: If f is a homomorphism of G o to G with kerel K, the G/ K is isomorphic to G 13) Cayley s theorem: Every fiite group of order is isomorphic to a permutatio group of degree 14) Rig: A algebraic system S,, is called a rig if the biary operatios ad o S satisfy the followig properties (i) S, is a abelia group (ii) S, is a semi group (iii) The operatio is distributive over Example: The set of all itegers Z, ad the set of all ratioal umbers R are rigs uder the usual additio ad usual multiplicatio 15) Commutative rig: 16) Itegral domai: A commutative rig without zero divisor is called Itegral domai Example: (i) R,, is a itegral domai, sice a, b R such that a 0, b 0 the 0 ab (ii),, Z is ot a itegral domai, Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 8

9 because 2,3 Z10 ad Therefore 2 ad 5 are zero divisors 17) Field: A commutative reig S,, which has more tha oe elemet such that every ozero elemet of S has a multiplicative iverse i S is called a field Example: The rig of ratioal umbers Q,, is a field sice it is a commutative rig ad each o-zero elemet is iversible Uit V (Lattices ad Boolea algebra) 1) Partially ordered set (Poset): A relatio R o a set A is called a partial order relatio, if R is reflexive, atisymmetric ad trasitive The set A together with partial order relatio R is called partially ordered set or poset Example: The greater tha or equal to relatio is a partial orderig o the set of itegers Z 2) Lattice: A lattice is a partially ordered set L, i which every pair of elemets a, b L has a glb ad lub 3) Sub-lattice: 4) Geeral formula: glb a, b a* b a b i) ii) l ub a, b a b a b iii) iv) 5) Properties: Name of Law a * b a & a * b b a b a * b a If a b b Primal form Idempotet law a * a a Commutative law & a * b b* a a b a a b b Dual form a a a a b b a Associative law a * b* c a * b* c a b c a b c Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 9

10 Distributive law p* q r p* q p* r p q* r p q* p r Absorptio law a* a b a * a a b a Complemet a* a 0 aa 1 Demorga s law * * a b a b a b a b Double Negatio law p p 6) Complemeted Lattices: A Lattice L,*, is said to be complemeted if for ay a L, there exist a L, such that a* a 0ad aa 1 7) Demorga s laws: Let L,*, be the complemeted lattice, the * a b a b * a b a b ad 8) Complete Lattice: A lattice L,*, is complete if for all o-empty subsets of L, there exists a glb ad lub 9) Lattice Homomorphism: Let L,*, ad S,, be two lattices A mappig g : L S homomorphism if g( a * b) g( a) g( b) ad g( a b) g( a) g( b) 10) Modular Lattice: A lattice L,*, is said to be modular if for ay a, b, c L i) ii) * * a c a b* c a b * c a c a b c a b c 11) Chai i Lattice: Let L, be a Chai if is called lattices i) a b or a c ad ii) a b ad a c 12) Coditio for the algebraic lattice: A lattice L,*, is said to be algebraic if it satisfies Commutative Law, Associative Law, Absorptio Law ad Existece of Idempotet elemet 13) Isotoe property: Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 10

11 Let L,*, be a lattice The biary operatios * ad are said to possess isotoe b c a * b a * c property if a b a c 14) Boolea Algebra: A Boolea algebra is a lattice which is both complemeted ad distributive It is deoted by B,*, ---- All the Best ---- Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 11