Bm Session 5

7/23/2019 Bm Session 5

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MSc-ITSemester - I

Basic Mathematics



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Binary operation

• Binary Operations: A binary operation is

simply a rule for combining two objects of agiven type, to obtain another object of that type.

• Through elementary school and most of high

school, the objects are numbers, and the rule for

combining numbers is addition, subtraction,

multiplication or division.




• Binary operation on a set S. A binary

operation on a set S is a rule which

assigns to each ordered pair a,b of

elements in S a unique element

• c = ab.

.




• Closure: A set S is closed with respect toa binary operation if and only if every

image ab is in S for every a,b in S.




Elementary terms and notation• Set – a collection of objects – not otherwise defined in

naïve set theory

• Correspondence – can be one-to-one or many-to-one orone-to-many

• Common symbols

Belongs to – is a member of

For all

There exists (at least one)

Not equal




Common relationships and definitions

• Equality – relationship is an equality relationship if:

– Reflexive a = a

– Transitive a = b and b = c imply a = c

– Symmetric a = b implies b = a

– Objects do not need to be equal numerically to satisfyan equivalence relationship – example, similar

triangles




• Closure a,b S implies a b S

• Associativity a (b c) = (a b) c – can be writtena b c

• Identity e S such that a S e a = a, a e = a

• Inverse a S a’ S such that a’ a = e, a a’ = e

• Commutativity a,b S a b = b a

• Distributivity a(b + c) = ab + ac




Properties of Binary operations:

• Commutative operation: A binary operation

on a set S is called commutative if

xy = yx for all x,y in S.


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• Associative operation: A binaryoperation on a set S is called associative if

• (xy)z = x (yz) for all x,y,z in S.



• Distributive: Let S be a set on which two

operations ∙ and + are defined. The operation ∙ is

said to left distributive with respect to + if

a ∙(b + c ) = (a∙b) + (a∙c) for all a,b,c in S • and is said to be right distributive with respect to

+ if

(b + c)∙a = (b∙a) + (c∙a) for all a,b,c in S



• Existence of identity elements and

inverse elements:



• Identity element: A set S is said to have

an identity element with respect to a binary

operation on S if there exists an element e

in S with the property ex = xe = x for every

x in S.



• Inverse element: If a set S contains an

identity element e for the binary operation ,

then an element b S is an inverse of an

element a S with respect to if ab = ba = e .

• Note. There must be an identity element in

order for inverse elements to exist.



Field• A field is a set of two or more elements

F ={ , ,..} closed under two operations, +

(addition) and * (multiplication) with thefollowing properties

– F is an Abelian group under addition

– The set F −{0 } is an Abelian group under

multiplication, where 0 denotes the identityunder addition.

– The distributive law is satisfied:

( + +



Theorems :



• Theorem 1.

• A set S contains at most one identity for

the binary operation.



• An element e is called a left identity if

ea = a for every a in S.

• It is called a right identity ifae = a for every a in S.

• If a set contains both a left and a rightidentity, they are the same



• Theorem 2.

• An element of a set S can have at mostone inverse if the operation is associative



Theorem 3.

• Let a set S be closed with respect to an

associative binary operation. Then the

products formed from the factors

multiplied in that order, and with theparentheses placed in any positions

whatever, are equal to the general product



Operation Table



Interpretation

• We interpret this operation table in much the

same way that we would interpret an addition

table.• Using the operation symbol * as we would use +

to mean addition, the table shows us, among

other things, that



• Not A l l Operat ions Have the SamePropert ies



• Addition of numbers, for instance, is a

commutative operation -- meaning

that x+y= y+x for all numbers x and y .

• The operation on the set A defined by the

operation table above, however, is not

commutative, and there are several

instances of this lack of commutativity.



• Lack of commutativity



• For instance, since the table

shows that

• In general, commutativity is a property of

an operation, so it takes only one instance

of lack of commutativity to spoil that

property for the operation.



• It is easy to check whether an operation

defined by a table is commutative. Simply

draw the diagonal line from upper left to

lower right, and then look to see if thetable is symmetric about this line.



• In the illustration below, we see a lack of

symmetry: the table entries colored yellow

do not match, and the table entries colored

blue do not match




• Lack of Associativity




• Ex. Consider the operation defined on the set S= {1,2,3}

by the operation table below.

From the table, we see

2 (1 3)=2 3=2 but (2 1) 3=3 3=1

The associative law fails to hold in this groupoid(S, )

2

1

3

*1 2 3

1

2

3

1

3

2

3

2

1




Groupoid

• The groupoid is an algebraic structure on a set with abinary operator.

• The only restriction on the operator is closure (i.e.,applying the binary operator to two elements of a givenset S returns a value which is itself a member of S ).

• Associativity, commutativity, etc., are not required.

• A groupoid can be empty.• An associative groupoid is called a semigroup.




群胚 Groupoid

• A groupoid must satisfy

is closed under the rule of combination R

, R

R baR b,a




• A semigroup is a groupoid whose operation

satisfies the associative law.

(groupoid)

Semigroup半群

c bac baR c, b,a

R baR b,a




• A mathematical object defined for a set and a binary

operator in which the multiplication operation is

associative. No other restrictions are placed on a

semigroup; thus a semigroup need not have an identity

element and its elements need not have inverses withinthe semigroup.

• A semigroup is an associative groupoid.

• A semigroup with an identity is called a monoid.

• A semigroup can be empty.




Monoid• A monoid is a set that is closed under an associative

binary operation and has an identity element such that

for all• Note that unlike a group, its elements need not have

inverses. It can also be thought of as a semigroup with

an identity element.

• A monoid must contain at least one element. A monoidthat is commutative is, not surprisingly, known as a

commutative monoid.




• A semigroup having an identity elementfor the operation is called a monoid.

(groupoid)

(semigroup)

Monoid, R

aaeeaR a Re

e

R baR b,a

c bac baR c, b,a




Ex. Both the semigroups and are instancesof monoids

for each

The empty set is the identity element for the unionoperation.

for each

The universal set is the identity element for the

intersection operation.

) ,(S U

) ,(S U

A A A U A

A AU U A U A




Group

• A set is said to be a group "under" this operation.Elements A, B, C, ... with binary operation between Aand B denoted AB form a group if

1. Closure: If A and B are two elements in G, then the

product AB is also in G.2. Associativity: The defined multiplication is associative,

i.e., for all ,

3. Identity: There is an identity element I (a.k.a. 1, , or )such that for every element .

4. Inverse: There must be an inverse (a.k.a. reciprocal) ofeach element. Therefore, for each element A of G , theset contains an element such that

http://mathworld.wolfram.com/IdentityElement.html

http://mathworld.wolfram.com/IdentityElement.html




• A group G is a finite or infinite set of elements

together with a binary operation (called the

group operation) that together satisfy the four

fundamental properties of closure, associativity,the identity property, and the inverse property.

The operation with respect to which a group is

defined is often called the "group operation,"




Group: example

• A set of non-singular n n matrices of real

numbers, with matrix multiplication• Note; the operation does not have to be

commutative to be a Group.

• Example of non-group: a set of non-negative

integers, with +




群 Group

• A monoid which each element of has

an inverse is called a group

(groupoid)

(semigroup)(monoid)

, R R

R baR b,a

c bac baR c, b,a

aaeeaR a Re

eaaaaR aR a 1-1-1




Abelian group

• If the operation is commutative, the group

is an Abelian group.

– The set of m n real matrices, with + .

– The set of integers, with + .




• If is a group and ,then

Proof. all we need to show is that

from the uniqueness of the inverse of

we would conclude

a similar argument establishes that

, R Rba,-1-1-1 abb)(a

eb)(a )a(b )a(bb)(a -1-1-1-1

ba

-1-1-1 abb)(a

eaa

)a(ea

)a )b((ba )a(bb)(a

1-

1-

-1-1-1-1

eb)(a )a(b -1-1




Group theory• The study of groups is known as group theory.

• If there are a finite number of elements, the group is

called a finite group and the number of elements iscalled the group order of the group.

• A subset of a group that is closed under the groupoperation and the inverse operation is called a subgroup.

• Subgroups are also groups and many commonlyencountered groups are in fact special subgroups ofsome more general larger group.




Commutative

可交換性

group

1-1-1

monoid

semigroup

groupoid

eaaaa Ra Ra

aaeea Re Ra

cb)(ac)(ba Rcb,a,

Rba Rba,

abba Rba,Commutative

groupoid Commutative

semigroup

Commutative monoid

Commutative group




• Multiple choice questions:




Ques.2: If ea = a for every a in S, then e is

called

• right identity

• left identity

• right inverse

• left inverse




Ans: Left identity




Ques.3: There must be an identity element

in order for inverse elements to exist

• Always true

• False

• Depends upon the elements of the set

• None




Ans: Always true




Ques.4: An algebric structure (G,*),

satisfying only the closure property and

the associative law, is called

• Semigroup

• Monoid

• Group

• Groupoid




Ans: Semigroup




Ques.5: A monoid each of whose elements

is invertible, is called

• Semigroup

• Cyclic group

• Group

• Groupoid




Ans: Group




Ques.6: Let S be a set on which two

operations ∙ and + are defined. The

operation ∙ is said to left distributive with

respect to + if for all a,b,c in S• a ∙(b + c ) = (b + c)∙a

• (b + c)∙a = b + c + a

• a ∙(b + c ) = (a∙b) + (a∙c)

• (b + c)∙a = (b∙a) + (c∙a)




Ans: a ∙(b + c ) = (a∙b) + (a∙c)




Ques.7: A semigroup with an identity

element, is called

• Cyclic group

• Monoid

• Group

• Groupoid




Ans: Monoid




Ques.8: Which one of the following is true

• A group must contain at least one element

• A monoid must contain at least one

element

• A semigroup can be empty

• All are true




Ans: All are true




Ques.9: An associative groupoid is called a

• a. Cyclic group

• b. Monoid

• c. Group

• d. Semigroup




Ans: Semigroup




Ques.10: A subset of a group that is closed

under the group operation and the inverse

operation is called

• Cyclic group

• Subgroup

• Abelian group

• Semigroup




Ans: Subgroup



Thank You

Bm Session 5

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