Set Operations

The relative complement (also: set difference) of $A$ in $B$, denoted by as $B - A$ or $B \setminus A$ (

$B \setminus A$

), is the set of elements in $B$ that are not in $A$: $B - A = \;$$\{ x \mid x \in B \,$$\text{ and } \,$$x \notin A \}$. Examples:

• If $A = \{1,2,3\}$ and $B = \{1,2,3,4,5\}$, then $B - A = \{4,5\}$.
• If $A = \{a, b\}$ and $B = \{a, b, c, d\}$, then $B - A = \{c, d\}$.
• If $A = \{1,2\}$ and $B = \{1,2\}$, then $B - A = \emptyset$.

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The symmetric difference of two sets $A$ and $B$, denoted by $A \triangle B$ (

$A \triangle B$

) is the set of elements in either $A$ or $B$, but not in both: $A \triangle B = \;$$(A - B) \cup (B - A)$. Examples:

• If $A = \{1,2,3\}$ and $B = \{3,4,5\}$, then $A \triangle B = \{1,2,4,5\}$.
• If $A = \{a, b, c\}$ and $B = \{b, c, d, e\}$, then $A \triangle B = \{a, d, e\}$.
• If $A = \{1,2,3\}$ and $B = \{1,2,3\}$, then $A \triangle B = \emptyset$.