What does the set notation $A\setminus B$ mean?

Question

If $A$ and $B$ are two sets. What does the notation $A\setminus B$ denote for? I found this notation in the inclusion and exclusion principle.

Answer 1

Also, be aware that many texts write $A - B$ to indicate the difference of two sets.

Answer 2

It is almost certainly in that context (although slightly possibly but very unlikely not) set exclusion.

$A \setminus B = \{a\in A|a \not \in B\}$. Or in other words "all the elements in $A$ that specifically are not in the set $B$.

Some: identities:

1) $A\setminus B = A \cap B^c$ (you can use this as a definition if you want).

2) $A = (A \setminus B) \cup (A \cap B)$ and $(A\setminus B) \cap (A\cap B)=\emptyset$

3) If $U$ is the universal set $A^c = U \setminus A$.

4) The universal set, $U$, consists of four disjoint but exhaustive sets:

I) All the elements that are in $A$ but not in $B$: $A \cap B^c = A \setminus B$

II) All the elements that are in $A$ and in $B$: $A \cap B$.

III) All the elements that are in $B$ but not in $A$: $B \cap A^c = B \setminus A$

IV) All the elements that are not in $A$ nor in $B$: $(A \cup B)^c = A^c \cap B^C = U \setminus (A \cup B)$.

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Sometimes it is written $A - B$ but that is frowned upon (Because if $A, B \subset \mathbb R$ the notation $A - B$ often means $\{a-b| a\in A, b\in B\}$. But that is ambiguous and not always universally accepted.)

Also beware, in Algebra the notation $A/B$ often means something entirely different (I won't confuse you with details here). BUt irritatingly a few very bad texts will use $A/B$ as notation for $A\setminus B$.