Question

In $\mathbb{Z}$, the set of all integers, find the inverse of $\left( { - 7} \right)$ with respect to $*$ defined by $a*b = a + b + 7,\forall a,b \in \mathbb{Z}$.
A. $ - 14$
B. $7$
C. $14$
D. $ - 7$

Answer 1

Hint: A binary operation is defined in the question. You have to find the inverse of the element $\left( { - 7} \right)$ with respect to the operation. Find the identity element in the group $\left( {\mathbb{Z},*} \right)$ using its definition.

Formula Used:
If $e$ be the identity element in the group then $a*e = e*a = a$
If $c$ be the inverse of the element $a$ then $a*c = c*a = e$

Complete step by step solution:
The given binary operation is $*$ and it is defined by $a*b = a + b + 7,\forall a,b \in \mathbb{Z}$
Let $e$ be the identity element in the group $\left( {\mathbb{Z},*} \right)$ and $a \in \mathbb{Z}$ be an arbitrary element.
Then $a*e = e*a = a$
According to the definition of the binary operation $*$, we have
$a*e = a + e + 7 = a$ and $e*a = e + a + 7 = a$
$ \Rightarrow e + 7 = 0$
$ \Rightarrow e = - 7$
$\therefore \left( { - 7} \right)$ is the identity element in the group $\left( {\mathbb{Z},*} \right)$
Let $c$ be the inverse of the element $a$ in the group $\left( {\mathbb{Z},*} \right)$
Then according to the definition of inverse of an element, we have
$a*c = c*a = - 7$
According to the definition of the binary operation $*$, we have
$a*c = a + c + 7 = - 7$ and $c*a = c + a + 7 = - 7$
$ \Rightarrow a + c = - 7 - 7$
$ \Rightarrow a + c = - 14$
$ \Rightarrow c = - 14 - a$
Thus, inverse of an element $a$ in the group $\left( {\mathbb{Z},*} \right)$ is $\left( { - 14 - a} \right)$
So, inverse of $\left( { - 7} \right)$ is $ - 14 - \left( { - 7} \right) = - 14 + 7 = - 7$

Option ‘D’ is correct

Note: If $e$ be the identity element in the group then $a*e = e*a = a$. Hence find the inverse of an element using its definition. If $c$ be the inverse of the element $a$ then $a*c = c*a = e$. After that, find the inverse of $\left( { - 7} \right)$ by putting $a = - 7$. Remember that the identity element and inverse of an element are unique always.