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:

$$A-B=\{x\in A \ and \ x\notin B\}$$

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

• 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 $W = \{x \ | \ x \ \text{ odd and } x < 13\}$ 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:

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

Some properties of the set difference:

1. $A-A=\emptyset$
2. $A-\emptyset=\emptyset-A=A$
3. $A-B=A\cap B^c$
4. $A\subset B \Leftrightarrow A-B=\emptyset$
5. $A-(A-B)=A\cap B$
6. $A\cap(B-C)=(A\cap B)-(A\cap C)$