Question

In $Z$, the set of all integers, the inverse of $-7$ with respect to $*$ defined by $a * b = a + b + 7$ for all $a, b \in Z$ is

• A. -14
• B. 7
• C. 14
• D. -7

Answer 1

Answer: (D)

-7

Answer 2

$\\{Z , * \\}$ is a group
where $*$ is defined by $a * b = a + b + 7$
Now, $a * e = a \Rightarrow\, a + e + 7 = a$
$\Rightarrow \, e +7=0 \Rightarrow \, e=-7$
$\therefore \, 7$ is the identity element. Identity element has its own inverse.
$\therefore \, (-7)^{-1} = - 7$