Answer to Question #140737 in Abstract Algebra for Kenrick Lee

Question

Use the fact that 0-a= -a to justify that
-(-5) equals 5.

Answer 1

We have given the fact that

$$0+(-a)=-a$$

Claim:$-(-a)=a$

Proof: Note that We can write $0=a+(-a)=(-a)+a$

Now add additive inverse of $(-a)$ to the right, which is $-(-a)$ to the original equation, we get

$$0+(-a)+(-(-a))=-a+(-(-a))\\ \iff 0+0=-a+(-(-a))\iff0=-a+(-(-a))\\$$

Now add $a$ to the left in above last equation, thus

$$a+0=a=a+(-a)+(-(-a))\\ \iff a=0+(-(-a))=(-(-a))=-(-a)$$

Proved.

Now put $a=5$ in the above claim, we get the desired result.