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kirbykirby · Answer

Let $\large e$ be the additive identity. Thus, you need to find such that $\large x \oplus e=x$ and $\large e \oplus x = x$. You can see that -1 works since:

$\large x\oplus -1=x+(-1)+1=x$ and
$\large -1 \oplus x =-1+x+1=x$

I'm not sure what $-x$ is here though. 
$\large x\oplus -x=x+(-x)+1=1$
$\large c\otimes (-x)=c(-x)+c=-cx+c$
I don't see what's significant about $-x$ from these operations.

anonymous · Answer

can it be additive inverse maybe?

kirbykirby · Answer

If it was the additive inverse, , then adding x with -x should give the zero element (i.e the additive identity). Since the zero element here is -1, but $x \oplus -x =1 \ne -1$, I'm not too sure what -x is.

anonymous · Answer

can it be i/x

anonymous · Answer

1/x*

kirbykirby · Answer

It seems to me like $-(x+2)$ would be the additive inverse, since $x \oplus -(x+2) = x+(-(x+2))+1 = x-x-2+1=-1$

kirbykirby · Answer

and -1 is the zero element

kirbykirby · Answer

Of course in "regular" numbers using the regular addition and multiplication, -x would be an additive inverse. But your set S has a different definition of addition and multiplication, so the "usual" additive inverse and additive identity may not apply

anonymous · Answer

ya, i am still confused with -1 part. and how did you get additive idetity to be -1

kirbykirby · Answer

Just by any chance, is S a vector space?

anonymous · Answer

yes

kirbykirby · Answer

Hm.. actually it seems strange to me that this would be a vector space :( One part of the requirements of a vector space is that $1\vec{x}=\vec{x}$.
Now in your vector space, $c \otimes x = cx+c$, now with c=1:
$1 \otimes x = 1x+1=x+1 \ne x$

anonymous · Answer

question says show that S is a vector space. So I am not proving it but assuming it is

kirbykirby · Answer

Yes I "suppose" you can assume it, but it doesn't seem to agree with all the properties of a vector space :S which is quite strange...  Because I am assuming if they are asking what -x signifies, then it should be something significant.

anonymous · Answer

exactly, I am so confused right now. For the question before this one -x signifies 1/x

kirbykirby · Answer

what was the vector space defined in your previous question?

kirbykirby · Answer

Hm.. maybe I interpreted the question wrong.. maybe they are just using -x as a Notation to mean the additive inverse? (Maybe this is the standard in your course) I understood the question as finding what -x could represent.

kirbykirby · Answer

@perl

kirbykirby · Answer

@ganeshie8