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