Algebraic Structure
Definition
- A non empty set S is called algebric structure w.r.t binary operation * if (a*b) ∈ S ∀ a,b ∈ S.
- '*' is closure opertation on S
-
- (N , +) ✓ it is closed
- (N , .) ✓
- (N , -) ✗ 3-6 = -3 ∉ N → not algebric structure
- (N , /) ✗
- (Z , + , - , . , /) ✗
- (R , + , - , . , /) ✓
- (Q , /) ✗ 5/0 = ⧝ ∉ Q
- (Q* , /) ✓
Properties
Closure Property
- A set 'A' w.r.t operator '*' is said to satisfy closure property.
if ∀ a,b ∈ A
then a * b ∈ A
Algebraic Structure
- If a set 'A' w.r.t operator '*' satisfy closure property then it is called AS.
- (A, *)
Associative Property
- A set 'A' w.r.t operator '*' is said to satify Associative property
if ∀ a,b,c ∈ A
(a*b)*c = a*(b*c)
Semi Group
- If a algebraic structure satisfy associative property then it is called S.G.
Identity Property
- A set 'A' w.r.t operator '*' is said to satisfy identity property
if ∀ a ∈ A
there is an element 'e' such that
a * e = e * a = a
Monoid
- If a semi group satisfy Identify property then it is called Monoid.
- Example
- a + _____ = a ? (0)
- a * _____ = a ? (1)
Inverse Property
- A set 'A' w.r.t operator '*' is said to satisfy Inverse property
if ∀ a ∈ A
there is an element a-1 such that
a * a-1 = a-1 * a = e(identity) - Example
- 2 + ____ = 0(identity) ? (-2)
- 3 + ____ = 0(e) ? (-3)
- 3 * ____ = 1(e) ? (1/3)
Group
- If a Monoid satisfy Inverse property then it is called Group.
Commutative Property
- A set 'A' w.r.t operator '*' is said to satisfy commutative property
if ∀ a,b ∈ A
a * b = b * a
Abelian Group
- If a Group satisfy commutative property then it is called Abelian Group
