Discrete Mathematics by Meri Dedania Assistant Professor MCA department

1. Discrete Mathematics by MeriDedania Assistant Professor MCA department Atmiya Institute of Technology & Science YogidhamGurukul Rajkot

2. Group Theory

3. Definition of Group A Group < G ,  > is an algebraic system in which  on G satisfies four condition • Closure Property For all x , y  G x  y  G • Associative Property For all x , y , z  G x  (y  z) = (x  y)  z • Existence of Identity element There exists an element e  G such that for any a  G x  e = x = e  x • Existence of Inverse Element For every x  G ,there exists an element denoted by a-1  G such that x-1  x = x  x-1 = e

4. Definition of abelian Group A Group < G ,  >in which the operation  is commutative is called abelian Group i.e. a,b  G , a  b = b  a Example < Z , + > is Abelian Group < Q , + > is abelian Group

5. Properties of Group Theorem 1 : Let e be an identity element in group < G ,  > , Then e is unique Proof : • Let e and e` are two identity in G • e e` = e if e` is identity • e e` = e` if e is identity • since ee` is unique element in G • e = e`

6. Theorem 2 : Inverse of each element of a group < G ,  > is unique Proof : • Let a be any element of G and e the identity of G • Suppose b and c are two different inverse of a in G. • a  b = e = b  a (if b is an inverse of a) • a  c = e = c  a (if c is an inverse of a) • Now , b = b  e = b  ( a  c) = (b  a)  c = e  c = c Thus a has unique inverse

7. Theorem 3 : if a-1 is the inverse of an element a of group < G ,  > then (a-1)-1=a Proof : • Let e be the identity of Group < G ,  > • a-1 a = e • (a-1)-1 (a-1 a) = (a-1)-1 e • ((a-1)-1 a-1)  a = (a-1)-1 • e  a = (a-1)-1 • (a-1)-1 = a

8. Theorem 4 : If < G ,  > be a group then for any two elements a and b of < G ,  > prove that ( a  b )-1 = b-1  a-1 rule of reversal Proof : • Let a-1 and b-1 are inverse of a and b respectively and e be the identity • a  a-1 = e = a-1  a • b  b-1 = e = b-1  b • (a  b)  (b-1  a-1) = [(a  b)  b-1]  a-1 • = [a  (b  b-1)]  a-1 • = [a  e]  a-1 • = a  a-1 • = e • Similarly , (b-1  a-1)  (a  b) = e • This show that b-1 and a-1 is inverse of b and a • Hence , ( a  b )-1 = b-1  a-1

9. Cancellation Property : if a , b and cbe any three elements of a group < G , > then ab = ac  b = c left cancellation ba = ca  b = c right cancellation Proof : • Let a  G and also a-1  G • aa-1 = e = a-1a • where e is identity of G • Now , ab = ac • a-1(ab) = a-1 (ac) • (a-1 a) b = (a-1 a)c • e . b = e . c • b =c • similarly , ba = ca • b = c

10. Permutation Group Definition of Permutation A permutation is one to one mapping of non empty set P , say onto itself Example : Let S = {1,2,3} Then function f : S  S f(1) = 2 f(2) = 3 f(3) = 1 Then permutation P1 = P2 =

11. P3 = P4 = P5 = P6 = There are n! of pattern of expressing Permutation . So if Set has 3 elements then pattern of expressing permutation is 3! = 6

12. Equality of Permutations : • Let f and g be two permutations defined on a non empty set P. Then f = g if and only if f(x) = g(x) x  P • Example • Let S = {1,2,3,4} and let permutation f and g are equal or not.. f = g = • Let S = {1,2,3,4} and let permutation f and g are equal or not.. f = g =

13. Permutation Identity • An Identity permutation on S , denoted by I , is defined as I(a) = a a  S • For example : f = Note : In identity permutation the image of element is element itself

14. Composition of Permutation ( Product of Permutation) • Let f and g be two arbitrary permutations of like degree , given by, f = g = on non empty set A. Then the composition (or Product) of f and g is defined as Continue…

15. f  g =  = • Example Let P1 = P2= P3 = Check P1  (P2  P3) = (P1  P2)  P3

16. Inverse Permutation • Every permutation f on set P = {a1,a2,a3,…,an} Possesses a unique inverse permutation , denoted by f-1 thus if f = Then f-1 =

17. Cyclic Permutation • Let t1,t2,…..,tr be r distinct elements of the set P = {t1,t2,…., tn}.Then the permutation p : P  P is defined by p(t1) = t2 , p(t2) = t3,….,p(tr-1)= tr, p(tr)=t1 is called a cyclic permutation of length r. • Example : The permutation P = is written as (1,2) , (3,4,6) , (5).. The cycle (1,2) has length 2 , The cycle length 3,The cycle 1.

18. Definition of Cyclic Group • If there exists an element a  G for some group < G , > such that every element of G can be written as some power of a , that is an for some integer n. then a Group < G ,  > is said to be cyclic Group • Every Cyclic Group is abelian • Example for set A = { ,  ,  ,} and binary operation 

19. If < G ,  > is an abelian group , then for all a , b  G show that ( a  b )n = an  bn Solution ( a  b )n = an  bn ( a  b )n+1=an+1  bn+1 ( a  b )n+2=an+2  bn+2 Now , ( an  bn ) ( a  b ) = ( a  b )n+1 = ( an+1  bn+1 )  (bn  a )=(a  bn) By cancellation , similarly bn+1  a = a  bn+1 Again bn+1  a = b(bn  a) = b(abn) i.e., abn+1 = b(abn)

20. Show that in a Group < G ,  > , if for any a, b  G , ( a  b )2 = a2  b2, then <G , > must be abelian Solution : Let < G ,  > be a Group and let a , b  G ( a  b )2 = a2  b2  ( a  b )  ( a  b ) = ( a  a)  ( b  b ) • a  ( b  a)  b = a  ( a  b)  b • By left and right cancellation property • b  a = a  b • Thus we have a  b = b  a . a,b  G • Hence < G ,  >is an abelian Group

21. Show that if every element in a group is its own inverse , then the group must be abelian Solution : Let a , b  G • a  b  G (by closure property) Now, a-1 = a and b-1 = b • ( a  b)-1 = a  b Now, ( a  b)-1 = a  b • b-1  a-1 = a  b • b  a = a  b • Thus we have a  b = b  a , a,b  G • Hence < G ,  >is an abelian Group

22. Write down Composition table for <Z7, +5> and <Z7*, 7> where Z7* = Z7 - {0}

23. Show that < {1} ,  > and < {1 , -1} ,  > are the only finite groups of nonzero real numbers under the operation of multiplication Solution:

24. Sub Group and Homomorphism • Definition of Sub Group : • Let < G ,  > be a Group and S  G, such that it satisfies the following condition: • e  G , Where e is the identity of < G ,  > • For any a  S , a-1  S • For a , b  S , a  b  S Then < S ,  > is called Sub Group of <G ,  > • For any group <G ,  > , <{e} ,  > and <G ,  > are Trivial Sub Groups of <G ,  >. • Let <Z-{0}, X> is a Group then <{1}, X> & <Z,X> are Trivial Sub group of <Z, X> • All other subgroups of <G ,  > are called Proper Subgroup • Let < {,-1,0,1} , X > is Proper subgroup of <Z , X >

25. Theorem : A subset S   of G is a subgroup of < G ,  > iff for any pair of elements a , b  S , a  b-1  S Proof : • Assume that S is a subgroup • if a , b  S then b-1  S and a  b-1  S • To prove the converse , let us assume that a , b  S and a  b-1  S for any pair a , b. • taking b = a , a  a-1 = e  S • From e , a, b  S e  a-1 = a-1  S • Similarly , b-1  S. • Finally , because a and b-1 are in S , we have a  b  S. • Hence , < S ,  > is a sub group of < G ,  >

26. Definition of Group Homomorphism • Let < G ,  > and < H ,  > be two Group. A mapping g : G  H is called a group homomorphism from < G ,  > to < H ,  > if for any a , b  G • g (a  b) = g(a)  g(b) • g(eG) = eH • g(a-1) = [g(a)]-1

27. Definition of Group Isomorphism • Let f : < G ,  >  < H ,  >.if f is one to one and onto. Then Group is called isomorphism • A homomorphism f : < G ,  >  < H ,  > is called an endomorphism • A Isomorphism f : < G ,  >  < H ,  > is called an automorphism

28. Definition Kernal of Homomorphism • Let < G ,  > and < H ,  > be two Groups and let f is homomorphism of G into H. The set of elements of G which are mapped into eH , the identity of H is called the kernal of the homomorphism and is denoted by Kf or Ker(f)

29. Theorem : The Kernal of homomorphism f : <G , >  < H ,  > is sub group of < G ,  > Proof : • Here f : < G ,  >  < H ,  > is homomorphism • Ker (f) = {x  G | f(x) = eH identity element of H} • k (f)   because eG  K(f) (f(eG)=eH) • let a , b  Kf • f (a) = eH & f(b) = eG • Now, f(ab-1) = f(a) . f(b-1) • = f(a) . [f(b)]-1 • = eH . eH-1 • = eH . eH • = eH • ab-1  Kf • Kf is a sub group of < G ,  >

30. Show that every interval of lattice is a sub lattice of a lattice Proof: • Let < L ,  > be a lattice and a , b  L • a , b  • a  a  b • a  [ a , b ] • also [a , b] = {x  L | a  x  b}  L • Let x , y  [ a , b ] • a  x  b , a  y  b • a  a  x  y  b  b • a  a  x  y  b  b • a  x  y  b • a  x y  b • x  y  [ a , b ] and x  y  [ a , b ] • [ a , b ] is sub lattice of the lattice < L ,  >

31. Draw Hasse Diagram of the poset {2,3,5,6,9,15,24,45},D . Find (i) Maximal and Minimal elements (ii) Greatest and Least members, if exist. (iii) Upper bound of {9,15} and l.u.b. of{9,15} , if exist. (iv) Lower bound of {15,24} and g.l.b. of{15,24} , if exist.

32. Definition Of Right Cosets • Let G be a Group and H is any sub Group of G. Let a be any element of G . Then set Ha = {ha : hH} is called a right coset of H in G generated by a. • Definition of Left Cosets • Let G be a Group and H is any sub Group of G. Let a be any element of G . Then set aH = {aH : hH} is called a right coset of H in G generated by a.

33. Lagrange’s Theorem : • The order of each sub group of a finite group G is a divisor of the order of G • Index in G : • The number of left cosets of H in G is called index of H in G. • Definition of Normal subgroup: • A sub group < H ,  > is sub group of < G ,  > is called a normal sub group if for any a  G , aH = Ha

34. Fins the Sub group of < Z12 , +12 > • show that <{1,4,13,16} , 17> is subgroup of < Z17* , 17 > • Show that every sub group of abelian group is normal • Let x  G and h  H • Xhx-1 = xx-1h = eh = h • x  G and h  H • xhx-1  H • i.e. xH = Hx • H is normal subgroup of G

35. Definition of Irreflexive • A relation R on a set A is irreflexive if aRa for  a  A, if (a,a)  R • For example • A = {1,2,3,4} • R = {<1,1>,<1,2>,<2,3>,<1,3>,<4,4>} • R = {<1,3>,<2,1>}