Question

In the set of integers under the operation * defined by a*b=a+b-1 the identity element is

Answer 1

Hint: In this type of question where we have to find the identity element. First of all, assume the identity element and then use the property that its multiplication with the element is the element itself. Then with the help of questions get one relation with the same identity element. Now, get the identity element after solving it.That is we have to solve \[a*e=a+e-1\] and \[a*e=a\] . Then, solve it and find the value of e.

Complete step by step answer:
Let the identity element be e. We know that the operation of an element with the identity element is the element itself.
\[\Rightarrow a*e=a..........eq(i)\]
But according to question,
\[\text{a*b=a+b-1}\]
So,
\[a*e=a+e-1\] ……..eq(ii)
We can say that eq(i) and eq(ii) are equal
\[\begin{align}
  & a+e-1=a \\
 & \Rightarrow e=1 \\
\end{align}\]

Hence, the identity element is 1.

Note: In this question,one can write \[a*e\] as \[ae\]. But this is wrong. As e is an identity element so it operates with a will be the element a only. So, we have to keep the property \[a*e=a\] in mind.