An algebra is (Ω , θ , Γ ) defined as :-
1. Ω : List of Symbol/Variable
2. θ : List of Operation/mapping from E1,E2,... ⊆ Ω* to F1,F2,... ⊆ Ω* [ '*' is Kleene closure
A* = ( ∅ ∪ A ∪ (A^2) ... )] .
3. Γ : List of properties that algebra follows
Sometimes we express an algebra by (Ω , θ) then mention the Γ i.e algebra A = (Ω , θ)| Γ
Now in abstract algebra deals with some mathematical models and definitions :-
Definition : A definition is a statement describing an object exactly as it is .
Set : An order-independent collection of well defined distinct objects .
Multiset : An order-independent collection of well defined [duplication is possible ] objects.
Tuple : An ordered list of elements .
Set operations : ∈ , ⊆ , ∪ , ∩ , ()' [ Complement ] ,2^() [Power Set]
1. x ∈ A means x is an element of set A
2. A ⊆ B means A is subset of B
3. A ∪ B = {x | x ∈ A ∨ x ∈ B }
4. A ∩ B = {x | x ∈ A ∧ x ∈ B }
5. A' = {x | x ∉ A}
6. 2^A = {x | x ⊆ A}
Product of Sets : A × B = {(x,y)| x ∈ A ∧ y ∈ B } # of element = |A|.|B|
Relation : R ⊆ A × B # of relation 2^|A|.|B|
Mapping : φ | A→ B is representing a map from A to B i.e related elements of A to elements of B
Function : φ | A→ B is representing a function from A to B where for one element x of A function
φ map to y i.e. only one elements of B .
Category of functions :
Let a function φ | A→ B
1. one-one : ∀ x ∈ A φ(x) = φ(y) ⇒ x = y # of one-one functions |B|!/(|B| - |A|)!
2. onto : ∀ y ∈ B y = φ(x) for some x ∈ A # of onto functions |B|^|A|
3. bijection : one-one onto function # of bijections |A|! = |B|!
Group Theory :
1. magma (set G, operator +) : ∀ x,y ∈ G (x+y) ∈ G
2. semigroup(magma G, operator +): ∀ x,y,z ∈ G (x+y)+z = x+ (y+z)
3. Monoid ( semigroup G, operator +): ∀ x ∈ G ∃ 0 ∈ G s.t x + 0 = 0 + x = x (o is the identity)
4. Group (semigroup G, operator +): ∀ x ∈ G ∃ (-x) ∈ G s.t x + (-x) = (-x) + x = 0 ( (-x) is the inverse
of x )
5. Ring (G ,+ , .) : 1. (G,+) is commutative Group and (G , .) is semigroup
2. ∀ x,y,z ∈ G (x+y)z = xz + yz and
z(x+y) = zx + zy (distributive law)
6. Field (G, + , .) : 1. (G , +) and (G - {0} , .) are abelian Groups
2. (G,+ , .) is a ring
Boolean Algebra ( B , +, . ,()') :
1. (B,+) and (B, .) are commutative Monoids with identity 0,1 respectively
2. x + x' = 1 and x.x' = 0
3. (B , + , .) and (B , . , +) are identity rings .
There are some theorems on boolean algebra :
1. x + x = x.x = x
2. x + 1 = 1 and x.0 = 0
3. (x')' = x
4. x + xy = x(x+y) = x (Absorption Law)
5. (xy)' = x' + y' and (x+y)' = x'y' (De'Morgan's Law)
6. Duality of '+' and '.' : If we change '+' and '.' in a valid boolean identity the new identity will also be
valid.
S is a Set then (2^S, ∪ , ∩ , ()') ; P is the set of logical atoms then (P,∨,∧ ,¬ ) are all boolean algebra .
One more thing we can have such Boolean Algebra say B as Kleene Algebra s.t we can find a partial order in B and arrange elements of B as (1, x1, x2 , ... ) where 1 ≤ x1 ≤ x2 ... where
x ≤ y ⇔ x + y = y
1. Ω : List of Symbol/Variable
2. θ : List of Operation/mapping from E1,E2,... ⊆ Ω* to F1,F2,... ⊆ Ω* [ '*' is Kleene closure
A* = ( ∅ ∪ A ∪ (A^2) ... )] .
3. Γ : List of properties that algebra follows
Sometimes we express an algebra by (Ω , θ) then mention the Γ i.e algebra A = (Ω , θ)| Γ
Now in abstract algebra deals with some mathematical models and definitions :-
Definition : A definition is a statement describing an object exactly as it is .
Set : An order-independent collection of well defined distinct objects .
Multiset : An order-independent collection of well defined [duplication is possible ] objects.
Tuple : An ordered list of elements .
Set operations : ∈ , ⊆ , ∪ , ∩ , ()' [ Complement ] ,2^() [Power Set]
1. x ∈ A means x is an element of set A
2. A ⊆ B means A is subset of B
3. A ∪ B = {x | x ∈ A ∨ x ∈ B }
4. A ∩ B = {x | x ∈ A ∧ x ∈ B }
5. A' = {x | x ∉ A}
6. 2^A = {x | x ⊆ A}
Product of Sets : A × B = {(x,y)| x ∈ A ∧ y ∈ B } # of element = |A|.|B|
Relation : R ⊆ A × B # of relation 2^|A|.|B|
Mapping : φ | A→ B is representing a map from A to B i.e related elements of A to elements of B
Function : φ | A→ B is representing a function from A to B where for one element x of A function
φ map to y i.e. only one elements of B .
Category of functions :
Let a function φ | A→ B
1. one-one : ∀ x ∈ A φ(x) = φ(y) ⇒ x = y # of one-one functions |B|!/(|B| - |A|)!
2. onto : ∀ y ∈ B y = φ(x) for some x ∈ A # of onto functions |B|^|A|
3. bijection : one-one onto function # of bijections |A|! = |B|!
Group Theory :
1. magma (set G, operator +) : ∀ x,y ∈ G (x+y) ∈ G
2. semigroup(magma G, operator +): ∀ x,y,z ∈ G (x+y)+z = x+ (y+z)
3. Monoid ( semigroup G, operator +): ∀ x ∈ G ∃ 0 ∈ G s.t x + 0 = 0 + x = x (o is the identity)
4. Group (semigroup G, operator +): ∀ x ∈ G ∃ (-x) ∈ G s.t x + (-x) = (-x) + x = 0 ( (-x) is the inverse
of x )
5. Ring (G ,+ , .) : 1. (G,+) is commutative Group and (G , .) is semigroup
2. ∀ x,y,z ∈ G (x+y)z = xz + yz and
z(x+y) = zx + zy (distributive law)
6. Field (G, + , .) : 1. (G , +) and (G - {0} , .) are abelian Groups
2. (G,+ , .) is a ring
Boolean Algebra ( B , +, . ,()') :
1. (B,+) and (B, .) are commutative Monoids with identity 0,1 respectively
2. x + x' = 1 and x.x' = 0
3. (B , + , .) and (B , . , +) are identity rings .
There are some theorems on boolean algebra :
1. x + x = x.x = x
2. x + 1 = 1 and x.0 = 0
3. (x')' = x
4. x + xy = x(x+y) = x (Absorption Law)
5. (xy)' = x' + y' and (x+y)' = x'y' (De'Morgan's Law)
6. Duality of '+' and '.' : If we change '+' and '.' in a valid boolean identity the new identity will also be
valid.
S is a Set then (2^S, ∪ , ∩ , ()') ; P is the set of logical atoms then (P,∨,∧ ,¬ ) are all boolean algebra .
One more thing we can have such Boolean Algebra say B as Kleene Algebra s.t we can find a partial order in B and arrange elements of B as (1, x1, x2 , ... ) where 1 ≤ x1 ≤ x2 ... where
x ≤ y ⇔ x + y = y
No comments:
Post a Comment