Backslash — Definition, Formula & Examples

Backslash is the symbol

\setminus

used to denote set difference. When you write

A \setminus B

, it means the set of all elements that are in

A

but not in

B

For sets $A$ and $B$ , the expression $A \setminus B$ denotes the relative complement of $B$ in $A$ , defined as $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$ .

Key Formula

A \setminus B = \{x : x \in A \text{ and } x \notin B\}

Where:

$A$ = The original set you start with
$B$ = The set whose elements you remove from A
$x$ = An arbitrary element being tested for membership

How It Works

Place the backslash symbol between two sets. The set on the left is your starting set, and the set on the right contains the elements you want to remove. Go through each element of the left set and keep it only if it does not appear in the right set. The result is a new set containing the "leftover" elements. Some textbooks use a minus sign (

A - B

) instead, but the backslash notation is more common in formal mathematics because it avoids confusion with subtraction of numbers.

Worked Example

Problem: Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6}. Find A \ B.

Identify shared elements: Check which elements of A also appear in B.

A \cap B = \{2, 4\}

Remove them from A: Keep every element of A that is not in B. Remove 2 and 4 from A.

A \setminus B = \{1, 3, 5\}

Answer:

A \setminus B = \{1, 3, 5\}

Why It Matters

Set difference appears frequently in probability, where you compute

P(A \setminus B)

to find the probability of event

A

occurring without event

B

. It is also central to proofs in discrete mathematics and is used in database queries to exclude specific records from results.

Common Mistakes

Mistake: Assuming A \ B equals B \ A.

Correction: Set difference is not commutative. A \ B removes elements of B from A, while B \ A removes elements of A from B. These generally produce different sets.