Notation for set minus

Question

Suppose $\mathcal{A} = \{a,b\}$ or $\mathcal{A} = \{a \}$. Is it ok to write $\mathcal{A} \setminus b$ to indicate \begin{cases} \mathcal{A} \setminus b & \text{if } b \in \mathcal{A} \\ \mathcal{A} & \text{otherwise} \end{cases} If not, what would be an appropriate notation for that?

Answer 1

Regardless of your intentions when you write $\mathcal A\setminus b$, to the reader $\mathcal A\setminus b$ indicates $\mathcal A\setminus b$ when $b\notin \mathcal A$ as well.

Be mindful of the widespread notation $A\setminus B=\{x\in A\,:\, x\notin B\}$, which carries on in all instances.

Answer 2

Some authors use that notation, but $\mathcal{A}\setminus\{b\}$ is more accurate.

Answer 3

You can write $A-\{b\}$ or $A\backslash \{b\}$ to denote the set of all elements in $A$ that are not in the set $\{b\}$. This is the set of all elements in $A$ that are not the element $b$. You can use this notation regardless of whether $b$ is an element of $A$ or not.