10th maths solutions
Click here to solution of Exercise 1.5
A nonempty 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 a∗b =b∗a
Quasi Group
Let G be a non
empty set then G satisfies closure law only
a,b ϵ G ⇨ a ∗ b ϵ G
SemiGroup
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

Closure 
Associative 
Identity 
Inverse 
Commutative 
Quasi 
Yes 
No 
No 
No 
No 
Semigroup 
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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