What does this "\" mathematics symbol mean?

Question

I am reading a paper which has a mathematical argument: $X \subset V \setminus \{0\}$.

Can anyone explain this statement to me? Especially the "$\setminus \{0\}$".

Many thanks in advance!

Answer 1

This is set difference (or relative complement): $A\setminus B$ is the set of all things in $A$ but not $B$. That is, $$A\setminus B=\{x: x\in A, x\not\in B\}.$$ The latex code is "\setminus."

For example:

• $A\setminus A=\emptyset$, for every $A$.

• In your example, $V\setminus \{0\}$ is the set of all nonzero vectors in $V$.

• Note that $A\setminus B$ makes sense even if $B\not\subseteq A$, e.g.:

  • $\{1,2\}\setminus \{2, 3\}=\{1\}$.

  • $\{1,2\}\setminus \{3\}=\{1, 2\}$.

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In particular, "$X\subset V\setminus \{0\}$" means "$X$ is a subset of $V\setminus \{0\}$," or more succinctly "$X$ is a set of nonzero vectors (in $V$)." I'm guessing here that "$V$" is a vector space.

Note that unfortunately the symbol "$\subset$" is sometimes used to denote proper subset and sometimes general subset; context will indicate whether $X$ is meant to be a set of nonzero vectors, or a set of some but not all nonzero vectors (almost certainly the former).

Answer 2

In general, $A \setminus B = \{x \in A \mid x \not\in B\}$. Perhaps you're more familiar with the notation $A-B$. $V\setminus \{0\}$ denotes the set of all non-zero elements of $V$.

Answer 3

The symbol $\setminus$ has several meanings, according to wiki, e.g., it sometimes also denotes double cosets $$ K\left\backslash G \right/ H $$ Of course, in the context $V\setminus \{0\}$ it is the set difference. For a similar discussion, see

What does "\" mean in math