If $X$ is a set containing element $0$, and we want the set that contains all elements in $X$ except $0$, we can write $X-\{0\}$.

If $X=\prod_{i=1}^n X_i$ and we want to denote the cartesian product $Y$ of all $X_i$, except some $X_j$, can we similarly write $Y=X/\{X_j\}$?

This seems wrong, because $X_j$ is not a subset of $X$. How do we write this?

Similarly, if we have an element $x\in X$, but we want the element $y\in Y$ that is a tuple equal to $x$, except with the $j^{th}$ element removed, (i.e. the element in $X_j$), then how do we write this?

(I think it's called the "projection" of $x$ onto $Y$? Is that true?)


1 Answer 1

  • $X\setminus\{0\}$ (or $X-\{0\}$) is commonly used as notation for the set that contains all elements of $X$ except $0$ (not $X/\{0\}$ as you suggest, which has the looks of a quotient).

  • You can write:$$Y=\prod_{i=1,i\neq j}^nX_i$$ to denote the product of all $X_i$ except $X_j$.

  • I suspect that $x=(x_1,\dots,x_n)$ and you want a notation of $(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)$. This is a notation on its own already. Sometimes it is denoted as $$\left(x_{1},\dots,\hat{x_{j}},\dots,x_{n}\right)$$but if you do that then feel obliged to inform the reader.

  • Formally $X=\prod_{i=1}^nX_i$ is a product equipped with projections $p_i:X\to X_i$ for every $i$ prescribed by $(x_1,\dots,x_n)\to x_i$, so label "projection" is already preoccupied in some sence. On the other hand $X$ can also be looked at as a product $Y\times X_j$ having projections $X\to Y$ and $X\to X_j$. In that context the function $X\to Y$ that leaves $x_j$ can be recognized as a projection.

  • $\begingroup$ Is there no established shorthand for $Y=\prod_{i=1,i\neq j}^nX_i$ if you've already defined $X=\prod_{i=1}^nX_i$, such as $Y=X/X_j$? Similarly, is there no shorthand for $y=(x_1,...,x_{i-1},x_{i+1},...,x_n)\in Y$ if you've already defined $x=(x_1,...,x_n)\in X$ and $X=\prod_{i=1}^nX_i$ and $Y=\prod_{i=1,i\neq j}^nX_i$? $\endgroup$
    – user56834
    May 8, 2019 at 8:44
  • $\begingroup$ Not that I know of. If you want shorthands then you must create them yourself and inform the reader. Further you should use backslash (not slash) in this context. $\endgroup$
    – drhab
    May 8, 2019 at 8:49
  • $\begingroup$ Isn't $Y=X/X_j$ (with slash and not backslash) the quotient, and exactly what I need? $\endgroup$
    – user56834
    May 8, 2019 at 8:52
  • 1
    $\begingroup$ @user56834: You can almost write $\prod_{i \ne j} X_i$ as a quotient of $\prod_{i=1}^n X_i$ by defining a relation $\sim$ by $(x_1, x_2, \dots, x_n) \sim (x'_1, x'_2, \dots, x'_n)$ if and only if $x_j = x'_j$. But then the quotient $\left( \prod_{i=1}^n X_i \right) / {\sim}$ is merely in (canonical) bijection with the set $\prod_{i \ne j} X_i$, not actually equal to it. The element $(x_1,\dots,x_{j-1},x_{j+1},\dots,x_n)$ corresponds with the equivalence class $[(x_1,\dots,x_{j-1},a,x_{j+1},\dots,x_n)]_{\sim}$, where $a$ is an arbitrary element of $X_j$. $\endgroup$ May 8, 2019 at 13:16
  • 1
    $\begingroup$ ...but as drhab wisely suggests, if you want to use any kind of special notation, you should tell the reader what you mean. You could certainly use $X/X_j$ if you wanted to, so long as you made it very clear exactly what you mean. $\endgroup$ May 8, 2019 at 13:17

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