$C/C$ is the same as {0} ? Basically, then I get confused about the difference between $X-C$ and $X/C - C/C$.
What is $X/C - C/C$, where $C \subset X$ are modules and $X/C$ is meant as quotient?
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abstract-algebra
algebraic-topology
1 Answers
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$X/C$ is a quotient module. Its elements are equivalence classes $$x + C = \{x + c : c \in C\}$$ of elements $x \in X$.
In particular, $C/C$ consists of only one class: the trivial class $0 + C$. It is thus canonically isomorphic to the trivial module $\{0\}$.
The notation $X - C$ is usually used to represent the set complement of $C$ in $X$, that is, $$X - C = \{x \in X : x \notin C\}.$$
Note that $X - C$ is a subset of $X$ while the elements of $X/C$ are totally different entities. Also, note that $X - C$ is a subset but not a submodule of $X$. [Do you see why?]
Lastly, if you have a submodule $M$ of $X$ containing $C$, it makes sense to form the quotient $M/C$, which naturally sits inside $X/C$. Therefore, we have a well-defined set complement $X/C - M/C$. As an exercise, can you describe its elements explicitly?
In particular, you can consider the set complement $X/C - C/C$. Again, can you describe its elements explicitly?