Homework Statement
The following are equivalent for S\subseteqR, S\neq\oslash, and R is a commutative ring with unity(multiplicative identity):
1. <S> is the ideal generated by S.
2. <S> = \bigcap(I Ideal in R, S\subseteqI) = J
3. <S> = {\sumrisi: is any integer from 1 to n, ri\inR \foralli...
I'm not really hesitant about 1-1. Sorry my syntax was kind of confusing there. I think I understand the math behind it, but it just doesn't make intuitive sense to me. We're essentially saying that the cardinality of an infinite set is the same as the cardinality of that same set minus one...
Yes that's what I meant by containing a sequence of distinct points. I think I got it now but I'm still not 100% convinced about the answer. Basically I want to say that there is a sequence of distinct points in X \ {x} where x->x1 and xn->xn+1 and for any other element y in X that y->y. That...
1. If X is an infinite set and x is in X, show that X ~ X \ {x}
A~B if there exists a one-to-one function from A onto B.
Attempt at a solution
I'm pretty much completely stumped on this problem. I know that since X is infinite then it contains a sequence of distinct points. So...