Definition

Set difference $A - B$

Algebra • Sets • Operations

Statement

For sets

A

and

B

, the difference

A - B

(also written

A ∖ B

) is the set of elements that belong to

A

but not to

B

. Its size is

n (A - B) = n (A) - n (A \cap B)

How to use it

1Strike out from $A$ exactly the members of $A \cap B$ ; nothing from $B$ is ever added.
2When only the counts are wanted, use $n (A - B) = n (A) - n (A \cap B)$ rather than listing elements.
3Read the order carefully: $A - B$ and $B - A$ are different sets and usually different sizes.

A = {2, 4, 6, 8}

and

B = {4, 8, 12}

, find

A - B

and

B - A

Step-by-step

1
$A \cap B = {4, 8}$
Why: The shared elements are the only ones either difference removes.
2
$A - B = {2, 6}, B - A = {12}$
Why: Remove the shared pair from each set in turn.

Answer

A - B = {2, 6}

and

B - A = {12}

Watch out

Set difference is not symmetric:

A - B \neq = B - A

in general, and

n (A - B) = n (B - A)

only when

n (A) = n (B)

Also known as

relative complement, set minus, A minus B, elements of A not in B