Sets differences

Let $A$ and $B$ be two sets. The set difference of $A$ and $B$ , denoted as $A - B$ , is the set of all the elements of $A$ that are not members of $B$ .

Let $A$ and $B$ be two sets. The set difference $A - B$ is:

$A - B = {x \in A a n d x \notin B}$

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Elements belonging to the set difference $A - B$ are those elements that belong to $A$ and do not belong to $B$ .

Example

If $A = {a, b, c, d}$ and $B = {b, d}$ , then $A - B$ és $A - B = {a, c}$ .
If $A = {a, b, c, d}$ and $B = {c, d, e, f}$ , then $A - B = {a, b}$ .
If $W = {x | x odd and x < 13}$ and $Z = {7, 8, 9, 10, 11, 12, 13}$ , then $W - Z = {1, 3, 5}$ and $Z - W = {8, 10, 12, 13}$ .

Note that the set difference operation is not a commutative operation and if $A$ , $B$ are two disjoint sets, then $A - B = A$ and $B - A = B$ .

The simetric difference of any two sets $A, B$ is defined as:

$A △ B = (A - B) \cup (B - A) = (A \cup B) - (B \cap A)$

Some properties of the set difference:

$A - A = \emptyset$
$A - \emptyset = \emptyset - A = A$
$A - B = A \cap B^{c}$
$A \subset B \Leftrightarrow A - B = \emptyset$
$A - (A - B) = A \cap B$
$A \cap (B - C) = (A \cap B) - (A \cap C)$