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Question

For the binary operation * on Z defined by a * b = a + b + 1 the identity element is
(a) 0
(b) −1
(c) 1
(d) 2

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Solution

(b) −1

Let e be the identity element in Z with respect to * such that

a * e=a=e * a, aZa * e=a and e * a=a, aZThen, a+e+1=a and e+a+1=a, aZe=-1Z, aZ

Thus, −1 is the identity element in Z with respect to *.

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