Difference

Definition 3.2.25   Let $A$ and $B$ be two sets. The difference $A \setminus B$ is the set of elements of $A$ which do not belong to $B$, namely

$$A \setminus B = \{ x \in A \; \vert \; x \notin B \}$$

If $A$ and $B$ are two subsets of the same set $E$, we have:

$$A \setminus B = A \cap B^C.$$

Figure 13: Difference

For example, take $A=\{ 1,3,4,5,7,9 \}$ and $B= \{ 1,2,4,6,7,8 \}$. Then $A \setminus B = \{ 5,9 \}$. Note that this operation has nothing in common with substraction of numbers.

Proposition 3.2.26   For any set $A$, the following properties hold:

1. $A \setminus A = \emptyset$;
2. $A \setminus \emptyset = A$.

Remark 3.2.27   If $A$ and $B$ be two subsets of the same set $E$, then we have:

$$A \setminus B = A \cap \Bar{B}.$$

Remark 3.2.28   For the difference of sets, neither the commutative law nor the associative law hold. Figures and give a counter-example to the associative law.

Figure 14: A counter-example to the associative law: first part.

Figure 15: A counter-example to the associative law: second part.

Here is another example: let $E$ be the set of all integers $x$ such that $\lvert x\rvert \leq 10$. Let $A=\{ x \in E ; \; -1 \leq x \leq 6 \}$, $B=\{ -5,-4,-2,0,1,2,3,4,5 \}$ and $C=\{ -9,-4,-2,0,2,4,6,8 \}$. Then we have:

$$A \setminus B = \{ -1,6 \}$$

$$(A \setminus B ) \setminus C = \{-1\}$$

$$B \setminus C = \{ -5, 1, 3,5 \}$$

$$A \setminus ( B \setminus C) = \{ -1, 0,2,4,6 \}.$$