\displaystyle \textbf{Question 1: }\text{Find the identity element in the set }I^+\text{ of all positive integers}
\displaystyle \text{defined by }a\ast b=a+b,\text{ for all }a,b\in I^+.
\displaystyle \text{Answer:}
\displaystyle \text{Let }e\text{ be the identity element in }I^+\text{ with respect to }\ast.
\displaystyle \text{Then }a\ast e=a=e\ast a,\text{ for all }a\in I^+.
\displaystyle a\ast e=a.
\displaystyle a+e=a.
\displaystyle \therefore e=0.
\displaystyle e\ast a=a.
\displaystyle e+a=a.
\displaystyle \therefore e=0.
\displaystyle \text{But }0\notin I^+.
\displaystyle \therefore \text{ there is no identity element in }I^+\text{ with respect to }\ast.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Find the identity element in the set of all rational numbers except }
\displaystyle -1 \ \text{with respect to }\ast\text{ defined by }a\ast b=a+b+ab.
\displaystyle \text{Answer:}
\displaystyle \text{Let }e\in Q-\{-1\}\text{ be the identity element with respect to }\ast.
\displaystyle \text{Then }a\ast e=a=e\ast a,\text{ for all }a\in Q-\{-1\}.
\displaystyle a\ast e=a.
\displaystyle a+e+ae=a.
\displaystyle e+ae=0.
\displaystyle e(1+a)=0.
\displaystyle \text{Since }a\ne-1,\;1+a\ne0.
\displaystyle \therefore e=0.
\displaystyle e\ast a=a.
\displaystyle e+a+ea=a.
\displaystyle 0+a+0=a.
\displaystyle \therefore e=0\text{ satisfies both }a\ast e=a\text{ and }e\ast a=a.
\displaystyle \therefore 0\text{ is the identity element with respect to }\ast.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{If the binary operation }\ast\text{ on the set }Z\text{ is defined by}
\displaystyle a\ast b=a+b-5,\text{ then find the identity element with respect to }\ast.\qquad\text{[CBSE 2012]}
\displaystyle \text{Answer:}
\displaystyle \text{Let }e\in Z\text{ be the identity element with respect to }\ast.
\displaystyle \text{Then }a\ast e=a=e\ast a,\text{ for all }a\in Z.
\displaystyle a\ast e=a.
\displaystyle a+e-5=a.
\displaystyle \therefore e=5.
\displaystyle e\ast a=a.
\displaystyle e+a-5=a.
\displaystyle \therefore e=5.
\displaystyle \therefore 5\text{ is the identity element with respect to }\ast.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{On the set }Z\text{ of integers, if the binary operation }\ast\text{ is defined by}
\displaystyle a\ast b=a+b+2,\text{ then find the identity element.}\qquad\text{[CBSE 2012]}
\displaystyle \text{Answer:}
\displaystyle \text{Let }e\in Z\text{ be the identity element with respect to }\ast.
\displaystyle \text{Then }a\ast e=a=e\ast a,\text{ for all }a\in Z.
\displaystyle a\ast e=a.
\displaystyle a+e+2=a.
\displaystyle \therefore e=-2.
\displaystyle e\ast a=a.
\displaystyle e+a+2=a.
\displaystyle \therefore e=-2.
\displaystyle \therefore -2\text{ is the identity element with respect to }\ast.
\displaystyle \\

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