Answer to Question #116062 in Abstract Algebra for Joshua

Question

Every element
x in mathbbZ has ____________ inverses with to respect to addition as a binary operation.
a.\\(-x\\)
b.\\(0\\)
c.\\(frac{1}{x}\\)
d.\\(x\\)

Answer 1

First we will find the identity of $Z$ with respect to addition.

There exist $0 \in Z$ called identity of $Z$ such that $x+0=x$. So, $e=0$ is identity of $Z$ with respect to addition.

Now, let $x$ be any element of $Z$, then $y$ be the inverse of $x$ if $x+y=y+x = 0$

$\implies y = -x$ and since $x\in z \implies y=-x \in Z$.

So, $-x$ is the inverse of $x$.

Correct option is (a) $-x$ .