2. Following are the notations and definitions to represent the set of
numbers:
4. Binary Operations (Compositions)
Let S be a non-empty set. The mapping which associates each ordered
pair of the elements of S to a unique element of S, denoted by is called a
binary operation or a binary composition on S.
That is,
An operation on a non-empty set S is a binary operation if and only if
(S is closed under binary operation)
5. Examples:
1. The usual addition (+) and multiplication are binary operations on the set
of
(a) N of Natural Numbers (b) Z of integers
(c) Q of Rational Numbers (d) R of Real numbers
(e) C of Complex Numbers.
2. The usual subtraction is not a binary operation on the set N of natural
numbers. For,
6. Laws of Binary Operation
A binary operation on a non – empty set S
is said to be commutative if
is said to be associative if
is said to have an identity element if
is said to have an inverse element if
7. 4. If the binary operation on the set of real numbers is defined by . Show
thatis commutative and associative. Also find the identity element and the
inverse of.
Solution:
By definition
Now,
(Since usual addition is commutative)
11. Algebraic Structures:
A non – empty set together with one or more binary
compositions, satisfying certain axioms on is called an algebraic
structure or an algebraic system.
Examples:
12. Group:
A non – empty set together with a binary operation on G is
called a Group, if it satisfies the following axioms:
(1) G is closed under binary operation
(2) G is associative
(3) Existence of Identity: There exist an element such that
13. (4) Existence of Inverse: For every element there exist an element such that
is called inverse element of
Semi – Group:
A non – empty set together with a binary operation on G is called a Semi –
Group, if it satisfies the closure property and associativity.
Abelian Group:
A group is called an Abelian Group or a Commutative Group if
14. Finite Group:
If the number of elements in the group is finite then it is called Finite
Group; otherwise it is called an Infinite Group.
Order of the Group:
The number of elements in a group G is called the Order of the Group and
is denoted by.
Note:
1. An abelian group is a group with commutative law (or axiom)
2. A Group is a semigroup with identity and inverse axioms
15. Property 1: The identity element of a group G is unique.
Property 2: The inverse of an element of a group G is unique.
Property 3: If is an element of a group then.
Property 4: (Reversal Law of Inverse)
Let be a group and . Then.
Property 5: In a group for all.
(1) (Left Cancellation Law)
(2) (Right Cancellation Law)
16. 2. Show that where is the set of all positive rational numbers is a group; where
is defined by .
Solution:
By data
(i) Closure Property:
If .
Thus,
20. Hence is a group.
Further
(v) Commutative Property:
(
Hence is an abelian group.
21. 3. Show that the set is not a group under multiplication.
Solution:
(i) Closure Property:
If .
Thus,
(ii) Associative Property:
22. (iii) Existence of Identity:
(iv) Existence of Inverse:
But
For example,
Hence has no inverse under multiplication on
Hence is not a group.
23. 6. Show that the set of all fourth roots of unity form a group under usual
multiplication.
Solution:
Let be the set of all fourth roots of unity.
Now, the Cayley table is
24. (i) From the table it follows that is closed under multiplication.
(ii) Associative Law: Consider
So associative law holds good for any three elements in the set.
(iii) From the table it is clear that the identity element is
25. (iv) Existence of inverse: From the table it follows that
The inverse of is
The inverse of is
The inverse of is and
The inverse of is
(v) Usual multiplication is commutative and is clear from the table ( as
the table is symmetric about the principal diagonal).
Hence is an abelian group.
26. 8. In the group of non zero real numbers the binary operation is defined by . Solve.
Solution:
By definition, .
Given
27. Modular System
There are two special types of binary operations on the set of
integers. They are the following
(i) Addition modulo
(ii) Multiplication modulo
28. (i) Addition modulo :
Let be any two integers and let be a fixed positive integer. The binary
composition addition modulo is denoted and defined as
the .
Examples,
(1)
(2)
29. (ii) Multiplication modulo :
Let be any two integers and let be a fixed positive integer. The binary
composition multiplication modulo is denoted and defined as
the .
Examples,
(1)
(2)
30. The algebraic systems under the binary operations addition modulo and
multiplication modulo are called modular systems
Note:
is the set of all non-negative remainders obtained by dividing any
integer by . That is
Examples:
(1) If , then
(2) If , then
32. 2. Write the multiplication modulo 10 table for the set
Solution:
33. 3. Show that where is an abelian group.
Solution:
Given
34. (i) From the table it follows thatis closed under addition modulo 6.
(ii) Associative Law: Consider
So associative law holds good for any three elements in the set.
35. (iii) From the table it is clear that the identity element is
(iv) Existence of inverse: From the table it follows that
The inverse of is
The inverse of is
The inverse of is
The inverse of is
The inverse of is
The inverse of is
36. (v) For any two elements leaves the same remainder when divided
by 6. Thus
is commutative.
Hence is an abelian group.
37. 5. In the group , find the element
Solution:
Given
Here identity element is
From the table is .
Hence (from the table)
38. 6. In the group under multiplication modulo 7, find .
Solution:
Given
Here identity element is 1
From the table is .
39. SUBGROUP:
A subset of a group is said to be a subgroup of if is a group by itself under
the same binary operation induced by .
Note:
1. If is a subgroup of a group , then .
2. A group will have at least two subgroup
(i) , where is the identity element of
(ii)
They are called trivial (or improper) subgroups of
40. Other subgroups are called non-trivial (or proper) subgroups of .
Examples:
(1) The group of integers is a subgroup of the group .
(2) The group of non – zero rational numbers is a subgroup of
the group of non – zero real numbers.
(3) is a subgroup of the group
41. Theorem 1:
A non – empty subset of a group is a subgroup of if and only if
(i)
(ii)
Theorem 2:
A non – empty subset of a group is a subgroup of if and only if
42. Problems
1. Show that the set of even integers including is a subgroup of additive group of the
set of integers.
Solution:
Let be the set of even integers including .
Clearly
Let
Then
