# Question #1ba91

Mar 13, 2017

The Inverse of $2$, denoted by, ${2}^{-} 1$, under the

operation $\ast$ is $12 \in \mathbb{Z} .$

#### Explanation:

We assume that the Binary Operation $\ast$ is a function,

$\ast : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} : a \ast b = a + b - 7 , \forall a , b \in \mathbb{Z} .$

Now, to find the inverse of $2$ w.r.t. $\ast$, we have to first find the

Identity Element , say, $e$ for $\ast$.

By the Defn. of $e$, then, $a \ast e = e \ast a = a , \forall a .$

$a \ast e = a \Rightarrow a + e - 7 = a \Rightarrow e = 7.$

Also, $7 \ast a = a \Rightarrow 7 + a - 7 = a .$

So, $7$ is the Identity for $\ast$.

Now, suppose that, $x$ is an Inverse of $2 \text{ under } \ast$.

Then, by Defn. of an inverse, we must have,

$x \ast 2 = 2 \ast x = e = 7.$

$x \ast 2 = 7 \Rightarrow x + 2 - 7 = 7 \Rightarrow x = 12.$

We also have, $2 \ast 12 = 2 + 12 - 7 = 7 = e .$

Thus, $12 \ast 2 = 2 \ast 12 = e \left(= 7\right) .$

Therefore, the Inverse of $2$, denoted by, ${2}^{-} 1$, under the

operation $\ast$ is $12 \in \mathbb{Z} .$

N.B.: In fact, it is easy to see that, $\forall a \in \mathbb{Z} , {a}^{-} 1 = 14 - a .$

Enjoy Maths.!