Set Difference

What is Set Difference?

You have tasks. Your friend already did some of them.

What’s left? That’s set difference — what’s in A but not in B.

Symbol: $\setminus$ or $-$

$A \setminus B$

This reads: “A minus B”

$A = \{1, 2, 3, 4\}$ $B = \{3, 4, 5\}$

Find $A \setminus B$ :

Go through A — is it also in B? If so, remove it.

$A \setminus B = \{1, 2\}$

Keep what’s only in A.

$A$	$B$	$A \setminus B$	Why?
$\{a, b, c\}$	$\{b, c, d\}$	$\{a\}$	Only a isn’t in B
$\{1, 2, 3\}$	$\{4, 5\}$	$\{1, 2, 3\}$	Nothing to remove
$\{1, 2\}$	$\{1, 2, 3\}$	$\emptyset$	Everything removed
$\{1, 2, 3\}$	$\emptyset$	$\{1, 2, 3\}$	Empty set removes nothing

Unlike union and intersection, order matters here.

$A \setminus B \neq B \setminus A$

Example:

$A = \{1, 2, 3\}, \quad B = \{3, 4, 5\}$

Different results. Think of it like subtraction: $5 - 3 \neq 3 - 5$ .

Difference with itself:

$A \setminus A = \emptyset$

Remove everything, nothing left.

Difference with empty set:

$A \setminus \emptyset = A$

Nothing to remove.

Empty set difference:

$\emptyset \setminus A = \emptyset$

Start with nothing, end with nothing.

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

This reads: “The set of all $x$ such that $x$ is in A and $x$ is not in B.”

Set difference asks: is it only in A?