### 10th maths solutions

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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 t he exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G 4.The existence of Inverse there exists a ʹϵ G such that a ʹ∗a = a∗ a ʹ, ⩝ 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 Semi-Group Let G be a non empty set then G satisfies closure law and associative law only c losure: a, b ϵ G ⇨ a ∗ b ϵ G a ssociative: a, b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c) Monoid Let G be a non empty set then G satis