A binary operation * is a set A is a function from A × A to A. If * is a binary operation in a set A then than for the * image of the ordered pair (a, b) ∈ A×A, we write a*b.
For example:
Addition + is a binary operation in the set of natural number N, integer Z and real number R.
Multiplication is a binary operation in N, Q, Z, R and C.
General properties
Propery-1: Let A be any set. A binary operation * A × A →A is commutative if for every a, a, b ∈A.
a* b = b * a
Property-2:Let A be a non-empty set. A binary operation *; A × A →A is associative if
a* b) * c = a * (b * c) for every a, b, c ∈A.
Property-3: Let * be a binary operation on a non-empty set A. If there exists an element e ∈A such that e * a = a * e = a for every a ∈A, then we call the element e as identity with respect to * in A.
Property-4: Let * be a binary operation on a non-empty set A and e be the identity element in A with respect the operation *. An element a ∈A is invertible if there exists an element b ∈A such that
a* b = b * a = e
In which case a andb are inverses of each other. For the operation * if b is the inverses of a ∈A then we can write b =
Cancellation laws
We denote a bin. operation by * in a set A, is to satisfy.
(i) Left cancellation law if for all a, b, c ∈A,
a* b = a * c ⇒b = c
(ii) Right cancellation law if for all a, b, c ∈A
b* a = c * a ⇒b = c
Algebraic system
If A is a set and * is a bin. operation on A, then we call (A, *) an algebraic structure.
Example: Let R be the set of real numbers, then (R, +) is an algebraic structure.
Example: If N denotes the set of natural numbers then (N, +) is an algebraic structure.
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