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Consider an algebraic system (G, *), where G is the set of all non-zero real numbers and * is a binary operation defined bya*b=(a*b)/4Show that (G, *) is an abelian group |
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Answer» TO PROVE The set G is the set of all non-zero real numbers forms an abelian GROUP under the operation * defined by PROOF 1. CHECKING FOR CLOSURE PROPERTY So * is closed 2. CHECKING FOR ASSOCIATIVE PROPERTY Then So a * ( b * c ) = ( a * b ) * c So * is associative 3. EXISTENCE OF IDENTITY ELEMENT Let a ∈ G Let e be the identity element Then e*a= a*e= a So 4 is the identity element 4. EXISTENCE OF INVERSE ELEMENT Let a ∈ G Let there exists b ∈ G such that a*b= b*a= e So G is a group CHECKING FOR COMMUTATIVE PROPERTY Let a, b ∈ G Now So a * b = b * a So ( G , * ) is commutative group Hence proved ━━━━━━━━━━━━━━━━ Learn more from Brainly :- 1. show me the set G = { x+y√3:x,y=Q} is a group w.r.t addition 2. Show that the set Q+ of all positive rational numbers forms an abelian group under the operation * defined by a*b= 1/2(... |
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