Question

In $Z,$ the set of all integers, the inverse of $-7w.r.t.*$defined by $a*b=a+b+7,\forall a,b\in Z$ is

• A. $-14$
• B. $7$
• C. $14$
• D. $-7$

Answer 1

The correct option is D

$-7$

Let us consider the identity element $k$ for any element $a\in Z.$

Now,

$$\begin{gathered} a*k=a \\ a+k+7=a[\because a*b=a+b+7] \\ \therefore k=-7 \end{gathered}$$

So, the identity element is $-7.$

For the inverse element,

$$\begin{gathered} \Rightarrow a*a^{-1}=-7 \\ \Rightarrow a+a^{-1}+7=-7[\because a*b=a+b+7] \\ \Rightarrow a+a^{-1}=-14 \\ \Rightarrow a^{-1}=-14-a \\ \Rightarrow a^{-1}=-14+7[\because a=-7] \\ \Rightarrow a^{-1}=-7 \end{gathered}$$

Hence, the inverse of $-7$ is $-7.$