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- 1. DISCRETE STRUCTURE Presented to:- Mrs. Shashi PrabhaBy:-Kumar Siddarth Bansal (100101114) GroupMansi Mahajan (100101126) Semi GroupAnadi Vats (100101030) MonoidAshwin Soman (100101056 ) Permutation groupJishnu V. Nair (100101100) homomorphism and isomorphism
- 2. GROUP(G,*) be an algebraic structure where * is binary operation, then (G,*) is called a group if following conditions are satisfied: 1.Closure law: The binary * is a closed operation i.e. a*b є G for all a,b є G. 2.Associative law: The binary operation * is an associative operation i.e. a*(b*c)=(a*b)*c for all a,b,c є G. 3.Identity element: There exist an identity element i.e. for some e є S, e * a=a*e,a є G.4.Inverse law: For each a in G, there exist an element a′ (inverse of a) in G such that a*a′=a′*a=e.
- 3. EXAMPLESConsider three colored blocks (red, green, and blue), initially placed in the order RGB.Let a be the operation "swap the first block and the second block", and b be the operation"swap the second block and the third block".We can write xy for the operation "first do y, then do x"; so that ab is the operation RGB→ RBG → BRG, which could be described as "move the first two blocks one position tothe right and put the third block into the first position". If we write e for "leave the blocksas they are" (the identity operation), then we can write the six permutations of the threeblocks as follows:e : RGB → RGBa : RGB → GRBb : RGB → RBGab : RGB → BRGba : RGB → GBRaba : RGB → BGR
- 4. SEMI GROUP An algebraic structure (S,*) is called a semigroup if the following conditions are satisfied:1.The binary operation * is a closed operation i.e. . a*b є S for all a,b є S (closure law) 2.The binary operation * is an associative operation i.e. a*(b*c)=(a*b)*c for all a,b,c є S. (associative law)
- 5. EXAMPLES1. (N,+),(N,*) (Z,+),(Z,*) (Q,+),(Q,*) (R,+),(R,*)are all semigroup where N,Z,Q,R respectively denote set of naturalnumbers, set of integers ,set of rational numbers, set of real numbers as; (N,+) is set of natural numbers a+(b+c)=(a+b)+c(associative law) 1+(2+3)=(1+2)+3 1+5=3+3 6+6Hence ,it holds associative law, and a,b,c є N, follows Clouser law
- 6. EXAMPLES2.We know that every group (G,*) is a semigroup.Thus G={1,2,3,4} is a groupunder multiplication moduls 5 is also the semi group. Proof:* 1 2 3 4 i)Closure law verified1 1 2 3 4 ii)Associative law verified i.e.(1*2)*3=1*(2*3).2 2 4 1 3 iii)Identity element = 13 3 1 4 2 thus, it is a semigroup.4 4 3 2 1
- 7. MONOID An algebraic structure (S,*) is called monoid if following conditions are satisfied: 1.The binary operation * is a closed operation. 2.The binary operation * is an associative operation3.There exist an identity element i.e. for some e є S, e * a=a * e=a for all a є S. Thus a monoid is a semi group (S,*) that has identity element
- 8. EXAMPLES1. For each operation * define below, determine whether it is a monoid or not:i)on N ,a*b=a2+b2a)Closure 2*5=4+25=29, 3*4=9+16=25…. i.e(a*b є G)b)Asociative (2*5)*3=2*(5*3) 29*3=2(25+9) (29)+(3)=(2)+(34) 850=4+1156 850 != 1160 not a monoid.
- 9. EXAMPLESii) on R,where a*b=ab/3a) Closure : 5*3=5*3/3=5 --- real 6*4=6*4/3=8 --- realb) Associative : 2*(5*3)=(2*5)*3 2*5=(2*5/3)*3 2*5/3=2*5*3/3*3 10/3=10/3
- 10. EXAMPLESiii)Identity ae=a ae=ae/3 2ae=0 e=0 Thus it is a monoid…
- 11. EXAMPLES2. Let * be the operation on set R of real numbers defined by a*b=a+b+2aba) Find 2*3,3*(-5), and 7* (½)b) Is (R,*) is a monoid ???c) Find identity elementd) Which element have inverse and what are they??? i) 2*3 = 2+3+2*2*3 =17 ii)3*(-5) = 3 – 5 + 2 * 3 * -5 = 3 – 5 -30 = -32
- 12. EXAMPLESiii) 7 * (½) = 7 + (½) + 2 * 7 * (½) = 14(½) =14.5b) Is (R,*) a monoid ??? a) closure: 2*3=17, 3*-5 = -32, 7 * (½) = 14.5 all are real no. i.e. ( a*b є G ) checked b) associative : (2*3)*4 = 2*(3*4) (2+3+2*2*3)*4 = 2*(3+4+2*3*4) (17*4) = (2*31)
- 13. EXAMPLES(17+4+2*17*4) = (2+31+2*31*2)21+136 = 33+124157=157Checkedc)Identity : a e=a identity a e = a+e+ae 0= ae+e+ae 0= 2ae+e e(2a+1)=0 e=0 identity element = 0
- 14. EXAMPLESc) Find inverse a a-1 = e [ but e = 0] a a-1 = 0 let a-1 = x ax = 0 [ax = a+x+2ax]2ax+x = -a x(2a+1)= -a x = (-a/2a+1) a-1 = [-a/2a+1]……. No inverse will be at a= (½)
- 15. PERMUTATION GROUPLet A be finite set .then a function f : A A is said to be permutation of Aifi) f is one-oneii) f is ontoi.e. A bijection from A to itself is called permutation of A.The number of distinct element in the finite set A is called the degree ofpermutation
- 16. EQUALITY OF TWO PERMUTATION Let f and g be two permutation on a set X.Then f=g if and only if f(x)=g(x) for all x in X. Example: f= g= Evidently f(1)=2=g(1) , f(2)=3=g(2) f(3)=4=g(3)Thus f(x)=g(x) for all xϵ{1,2,3} which implies that f=g
- 17. IDENTITY PERMUTATION If each element of a permutation be replaced by itself.then it is called theidentity permutation and is denoted by the symbol I.For example: I= Is an identity permutation.
- 18. PRODUCT OF PERMUTATIONThe product of two permutations f and g of same degree is denoted by fog or fg , meaning first perform f then perform g. f= g= Then fog =
- 19. INVERSE PERMUTATIONSince a permutation is one-one onto map and hence it is inversible , i.e, every permutation f on a set P={a1,a2,a3,….an} Has a unique inverse permutation denoted by f -1 Thus if f= Then f-1=
- 20. PROPERTIES 1. Closure property2. Associative property3. Existence of identity4. Existence of inverse
- 21. CYCLIC PERMUTATIONA permutation which replaces n objects cyclically is called a cyclic permutation of degree n. Let , P= We can simply write it S=(1 2 3 4)
- 22. EXAMPLES Let A = {1,2} then number of permution group = 2 Similarly if A={1,2,3} then no. of permutation group = 6The six permutations on written as permutations in cycle form are 1,(1 2),(1 3),(2 3),(1 2 3),(2 1 3)
- 23. EXAMPLE
- 24. HOMOMORPHISM AND ISOMORPHISM A homomorphism is a map between two groups which respects the groupstructure. More formally, let G and H be two group, and f a map from G to H (for every g∈G, f(g)∈H). Then f is a homomorphism if for every g 1,g2∈G,f(g1g2)=f(g1)f(g2). For example, if H<G, then the inclusion map i(h)=h∈G is a homomorphism. Another example is a homomorphism from Z to Z given by multiplication by 2, f(n)=2n. This map is a homomorphism since f(n+m)=2(n+m)=2n+2m=f(n)+f(m).
- 25. HOMOMORPHISM AND ISOMORPHISMA group isomorphism is a special type of group homomorphism. It is a mapping between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the respective group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. Isomorphic groups have the same properties and the same structure of their multiplication table. Let (G, *) and (H, #) be two groups, where "*" and "#" are the binary operations in G and H, respectively. A group isomorphism from (G, *) to (H, #) isa bijection from G to H, i.e. a bijective mapping f : G → H such that for all u and v in G one has f (u * v) = f (u) # f (v).Two groups (G, *) and (H, #) are isomorphic if an isomorphism between them exists. This is written: (G, *) (H, #) If H = G and the binary operations # and * coincide, the bijection is an automorphism.
- 26. EXAMPLESThe group of all real numbers with addition, (R,+), is isomorphic to the group of all positive real numbers with multiplication (R +, ): via the isomorphism f(x) = ex

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