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?)