If the binary operation * on the set Z is defined by a * b = a + … - Vidyadip

Question

If the binary operation * on the set Z is defined by a * b = a + b - 5, the find the identity element with respect to *.

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Answer

Let e be the identity element in Z with respect to * such that,
a * e = a = e * a, $\forall\ \text{a}\in\text{Z}$
a * e = a and e * a = a, $\forall\ \text{a}\in\text{Z}$
a + e - 5 = a and e + a - 5 = a, $\forall\ \text{a}\in\text{Z}$
e = 5, $\forall\ \text{a}\in\text{Z}$
Thus, 5 is the identity element in Z with respect to *.

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