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Title: Abstract Algebra Binary Operation
Description: This note consists of basics of Abstract Algebra that is Binary Operation, it's properties and theorems related to it. It is very helpful for the beginner that is 1st or 2nd year college student. This is my notes which I read and it helps me to understand the basics of Abstract Algebra. This note is not too long nor too short. It consists of the basics and necessary information related to Binary Operation which will definitely helps beginner.
Description: This note consists of basics of Abstract Algebra that is Binary Operation, it's properties and theorems related to it. It is very helpful for the beginner that is 1st or 2nd year college student. This is my notes which I read and it helps me to understand the basics of Abstract Algebra. This note is not too long nor too short. It consists of the basics and necessary information related to Binary Operation which will definitely helps beginner.
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Binary Operation :- Let A be a non-empty set
...
It is denoted by a*b
...
This operation is
necessarily a closed operation
...
Example: Addition of two integers is a binary
operation a+b ∈ Z ∀ a, b ∈ Z
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O = Set of odd integers is not closed since
O1 + O2 ∉ O
...
a*(b*c) = (a*b)*c
Commutative : a*b = b*a
Identity : e ∈ A is said to be identity
if e*a = a*e =a for every a ∈ A
...
Theorem 1 :- The identity element of any set A with
binary operation ‘*’ is unique
...
since, e is identity ⇒ a*e = a ∀ a ∈ A
⇒ e’*e = e’ [e’ ∈ A] ………… (1)
and since, e’ is identity ⇒ a*e’ = a ∀ a ∈ A
⇒ e’*a = a
⇒ e’*e = e [e∈ A]…………(2)
hence, from (1) and (2)
e’*e = e’ = e
therefore, identity element is unique
...
Theorem 2 :- For an associative binary operation ‘*’ on
a set with identity, an inversible element possesses
unique element
...
let e∈ A be identity
Hence, a*b1 = b1*a = e
and, a*b2 = b2*a = e
Now, b1*(a*b2) = b1*e = b1 ……………
...
Title: Abstract Algebra Binary Operation
Description: This note consists of basics of Abstract Algebra that is Binary Operation, it's properties and theorems related to it. It is very helpful for the beginner that is 1st or 2nd year college student. This is my notes which I read and it helps me to understand the basics of Abstract Algebra. This note is not too long nor too short. It consists of the basics and necessary information related to Binary Operation which will definitely helps beginner.
Description: This note consists of basics of Abstract Algebra that is Binary Operation, it's properties and theorems related to it. It is very helpful for the beginner that is 1st or 2nd year college student. This is my notes which I read and it helps me to understand the basics of Abstract Algebra. This note is not too long nor too short. It consists of the basics and necessary information related to Binary Operation which will definitely helps beginner.