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.
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• 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.
.
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• 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.
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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
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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
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• 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
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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.
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• 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
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• Existence of identity elements and
inverse elements:
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• 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.
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• 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.
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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:
( + +
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Theorems :
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• Theorem 1.
• A set S contains at most one identity for
the binary operation.
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• 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
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• Theorem 2.
• An element of a set S can have at mostone inverse if the operation is associative
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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
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Operation Table
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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
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• Not A l l Operat ions Have the SamePropert ies
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• 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.
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• Lack of commutativity
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• 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.
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• 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.
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• 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
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• Lack of Associativity
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• 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
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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.
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群胚 Groupoid
• A groupoid must satisfy
is closed under the rule of combination R
, R
R baR b,a
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• 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
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• 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.
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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.
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• 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
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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
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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
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• 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,"
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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 +
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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
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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 + .
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• 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
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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.
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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
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• Multiple choice questions:
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Ques.2: If ea = a for every a in S, then e is
called
• right identity
• left identity
• right inverse
• left inverse
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Ans: Left identity
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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
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Ans: Always true
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Ques.4: An algebric structure (G,*),
satisfying only the closure property and
the associative law, is called
• Semigroup
• Monoid
• Group
• Groupoid
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Ans: Semigroup
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Ques.5: A monoid each of whose elements
is invertible, is called
• Semigroup
• Cyclic group
• Group
• Groupoid
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Ans: Group
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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)
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Ans: a ∙(b + c ) = (a∙b) + (a∙c)
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Ques.7: A semigroup with an identity
element, is called
• Cyclic group
• Monoid
• Group
• Groupoid
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Ans: Monoid
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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
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Ans: All are true
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Ques.9: An associative groupoid is called a
• a. Cyclic group
• b. Monoid
• c. Group
• d. Semigroup
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Ans: Semigroup
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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
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Ans: Subgroup
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