Post on 15-Sep-2014
description
By:-
Kumar Siddarth Bansal (100101114) Group
Mansi Mahajan (100101126) Semi Group
Anadi Vats (100101030) Monoid
Ashwin Soman (100101056 ) Permutation group
Jishnu V. Nair (100101100) homomorphism and isomorphism
DISCRETE STRUCTURE
Presented to:- Mrs. Shashi
Prabha
(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.
GROUP
EXAMPLES
Consider 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 to the right and put the third block into the first position". If we write e for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows:e : RGB → RGBa : RGB → GRBb : RGB → RBGab : RGB → BRGba : RGB → GBRaba : RGB → BGR
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)
SEMI GROUP
1. (N,+),(N,*)
(Z,+),(Z,*)
(Q,+),(Q,*)
(R,+),(R,*)
are all semigroup where N,Z,Q,R respectively denote set of natural numbers, 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+6
Hence ,it holds associative law, and a,b,c є N, follows Clouser law
EXAMPLES
2.We know that every group (G,*) is a semigroup.Thus G={1,2,3,4} is a group under multiplication moduls 5 is also the semi group.
Proof:
i)Closure law verified
ii)Associative law verified i.e.(1*2)*3=1*(2*3).
iii)Identity element = 1
thus, it is a semigroup.
EXAMPLES
* 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
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 operation
3.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
MONOID
1. For each operation * define below, determine whether it is a monoid or not:
i)on N ,a*b=a2+b2
a)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.
EXAMPLES
ii) on R,where a*b=ab/3
a) Closure :
5*3=5*3/3=5 --- real
6*4=6*4/3=8 --- real
b) 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
EXAMPLES
iii)Identity
ae=a
ae=ae/3
2ae=0
e=0
Thus it is a monoid…
EXAMPLES
2. Let * be the operation on set R of real numbers defined by a*b=a+b+2ab
a) Find 2*3,3*(-5), and 7* (½)
b) Is (R,*) is a monoid ???
c) Find identity element
d) 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
EXAMPLES
iii) 7 * (½) = 7 + (½) + 2 * 7 * (½)
= 14(½) =14.5
b) 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)
EXAMPLES
(17+4+2*17*4) = (2+31+2*31*2)
21+136 = 33+124
157=157
Checked
c)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
EXAMPLES
c) 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)
EXAMPLES
Let A be finite set .then a function f : A A is said to be permutation of A if
i) f is one-one
ii) f is onto
i.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 of permutation
PERMUTATION GROUP
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
EQUALITY OF TWO PERMUTATION
If each element of a permutation be replaced by itself.then it is called the identity permutation and
is denoted by the symbol I.
For example:
I =
Is an identity permutation.
IDENTITY PERMUTATION
The 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 =
PRODUCT OF PERMUTATION
Since 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=
INVERSE PERMUTATION
1. Closure property2. Associative property3. Existence of identity4. Existence of inverse
PROPERTIES
A 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)
CYCLIC PERMUTATION
Let A = {1,2} then number of permution group = 2
Similarly if A={1,2,3} then no. of permutation group = 6
The six permutations on written as permutations in cycle form are
1,(1 2),(1 3),(2 3),(1 2 3),(2 1 3)
EXAMPLES
EXAMPLE
HOMOMORPHISM AND ISOMORPHISM
A homomorphism is a map between two groups which respects the group structure. 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 g1,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).
A 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, #) is a 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.
HOMOMORPHISM AND ISOMORPHISM
EXAMPLES
The 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