Ds

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