### 10th maths solutions

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Group (Defn) A non-empty set G with binary operation • or ∗ is said to be a 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 satisfies

Examples of Group and Abelian group 1. 1) Let G be the set of all-natural numbers with the binary operation + Is G form a group? Solution: - Given G = the set of all-natural numbers i.e, G = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} here binary operation is an addition (+), (G, +) is a group if G satisfy groups laws otherwise not group Note: See the last post to know groups law, groups defn www.mathsgeni.com/?m=1 1. Closure law we can take any two elements from given set G then apply given binary operation we get a new element also belongs to given set G. 2 ϵ G, 7 ϵ G 2 + 7 = 9, 9 also belongs to G. so G satisfies closure law 2. Associative law: a, b, c ϵ G ⇨ (a ∗ b) ∗ c = a ∗ (b ∗ c) 3, 4, 8 ϵ G (a ∗ b) ∗ c ⇨ (3 + 4) + 8 = 7 + 8 =15 a ∗ (b ∗ c) = 3 + (4 + 8) = 3 + 12 =15

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