# Further maths

A binary operation * is defined on the set R of real number by a * b = a + b + ab ( where a, b belong to R ). Calculate 5 * ( - 2 ) * and find the identity element e of R under the operation. Determine the inverse under * of a general element a belong to R and state which element has no inverse.

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1. Please work out the answer for me.

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2. a * b = a + b + ab
5 * ( - 2 ) = 5 + (-2) + (5)(-2) = -7

If e is the identity element the e*b = b
but e*b = e + b + eb
It follows that e + b + eb = b
e + eb = 0
e(1 + b) = 0
e = 0 or b = -1

check:
0*15 = 0 + 15 + 0(15) = 15
7*-1 = 7 - 1 -7 = -1

Since clearly a*b = b*a
the identity element is either 0 or -1

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