1. DISCRETE STRUCTURE
Presented to:-
Mrs. Shashi Prabha
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
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. 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 → RGB
a : RGB → GRB
b : RGB → RBG
ab : RGB → BRG
ba : RGB → GBR
aba : 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. EXAMPLES
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
6. 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:
* 1 2 3 4
i)Closure law verified
1 1 2 3 4
ii)Associative law verified i.e.(1*2)*3=1*(2*3).
2 2 4 1 3
iii)Identity element = 1
3 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 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
8. EXAMPLES
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.
9. 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
11. 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
12. 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)
14. 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)
15. PERMUTATION GROUP
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
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 the
identity permutation and
is denoted by the symbol I.
For example:
I=
Is an identity permutation.
18. PRODUCT OF 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 =
19. INVERSE 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=
20. PROPERTIES
1. Closure property
2. Associative property
3. Existence of identity
4. Existence of inverse
21. CYCLIC PERMUTATION
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)
22. EXAMPLES
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)
24. 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 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 ISOMORPHISM
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.
26. 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
