10th maths solutions

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Group Theory


Group (Defn)  

A non-empty set G with binary operation • or is said to be group if G satisfies groups laws,  group denoted by (G,) 

Groups laws  

      1.Closure law  

         a,b ϵ G ⇨ a ∗ b ϵ G

2.Associative law

     a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)

3.The existence of Identity

     the exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G

4.The existence of Inverse

     there exists aʹϵ G such that aʹ∗a = a∗aʹ, ⩝ a ϵ G.

Commutative Group or Abelian Group

Let G be a group with binary operation ∗  i.e (G,) be a group

then for every a,b ϵ G such that ab =ba

Quasi Group

      Let G be a non empty set then G satisfies closure law only

     a,b ϵ G ⇨ a ∗ b ϵ G

     Semi-Group

     Let G be a non empty set then G satisfies closure law and

     associative law only

     closure: a,b ϵ G ⇨ a ∗ b ϵ G

     associative: a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)

     Monoid

      Let G be a non empty set then G satisfies closure law ,

      associative law and identity existence only

      closure    : a,b ϵ G ⇨ a ∗ b ϵ G

      associative : a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)

      identity   :  the exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G

 

    Hints:

 Groups/Laws

Closure

Associative

Identity

Inverse

Commutative

Quasi

Yes

No

No

No

No

Semi-group

Yes

Yes

No

No

No

Monoid

Yes

Yes

Yes

No

No

Group

Yes

Yes

Yes

Yes

No

Abelian group

Yes

Yes

Yes

Yes

Yes

                        

 


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