492.36True or false with reasons.(i)The functione:N×N→N, defined bye(m,n)=mn, is anassociative operation.Solution.False.(ii)Every group is abelian.Solution.False.(iii)The set of all positive real numbers is a group under multiplication.Solution.True.(iv)The set of all positive real numbers is a group under addition.Solution.False.(v)For alla,b∈G, whereGis a group,aba−1b−1=1.Solution.False.(vi)Every permutation of the verticesv1, v2, v3of an equilateral trian-gleπ3is the restriction of a symmetry ofπ3.Solution.True.(vii)Every permutation of the verticesv1, v2, v3, v4of a squareπ4isthe restriction of a symmetry ofπ4.Solution.False.(viii)Ifa,b∈G, whereGis a group, then(ab)n=anbnfor alln∈N
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