Question about set notation regarding removal of elements
Consider the sets \begin{align*} \mathcal{A} &= \{1,2,3,5,7\}\\ \mathcal{B} &= \{2,4,6,8\}\\ \mathcal{R} &= \{2,5\} \end{align*}
I want to remove the elements in $\mathcal{R}$ from $\mathcal{A}$ and $\mathcal{B}$.
Does it make sense to write: \begin{align*} \mathcal{A}\backslash\mathcal{R} &= \{1,3,7\}\\ \mathcal{B}\backslash\mathcal{R} &= \{4,6,8\} \end{align*} even though $\mathcal{B}$ does not contain element $5$? I am asking because I have multiple sets that I want to remove a set of elements from, but I do not know if the sets contain the elements to remove. I cannot think of an alternative notation.
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$\begingroup$Your notation is perfect.
By definition, $X\setminus Y$ is the set of elements that belong to $X$ but not to $Y$.
It is irrelevant whether $Y$ is a subset of $X$ or not.
BTW, the correct TeX macro is \setminus, not \backslash. See also this.
This notation is usually shorthand for $B\cap R^c$. So this works fine if you imagine all your sets are subsets of some common set: in this case they're all subsets of $\mathbf{N}$ the natural numbers. It's also possible to write $B\setminus (R\cap B)$ if the situation worries you.
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