# Notation for a Cartesian product or tuple except one element?

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?)

• $$X\setminus\{0\}$$ (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.

• 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 $Y=\prod_{i=1,i\neq j}^nX_i$? May 8, 2019 at 8:44
• 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. May 8, 2019 at 8:49
• Isn't $Y=X/X_j$ (with slash and not backslash) the quotient, and exactly what I need? May 8, 2019 at 8:52
• @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 $X_j$. May 8, 2019 at 13:16
• ...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. May 8, 2019 at 13:17