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Consider a binary operation on Q -{1} define by a*b =a+b-ab (i) Find the identity element in Q-{1} (ii) Show that each a in Q-{1} has its invese

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Solution :(i) LET e be the identity element Then for all a in Q-{1} we have
`a*e=a rarr a+e-ae=a`
`rarr e(1-a=0 rarr e=0` [before a `ne` 1]
`therefore` 0 is the identity element
(II) Let a in Q-{1} be an arbitray element and let b be its INVERSE
Then `a*b=0 rarr a+b-ab=0 rarr ab-b=a`
`rarr b(a-1)=a rarr b=(a)/(a-1)`
Thus each a in Q-{1} has `(a)/(a-1)` as its inverse


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