Set Operations

We can perform several mixing and matching operations on sets to get new sets:

The intersection of two sets $A$ and $B$, denoted as $A \cap B$ (

$A \cap B$

), is the set of elements that are in both $A$ and $B$. Formally: $A \cap B = \;$$\{ x \mid x \in A \,$$\text{ and } x \in B \}$. Examples:

• If $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$, then $A \cap B = \{3,4\}$.
• If $A = \{a, b, c\}$ and $B = \{c, d, e\}$, then $A \cap B = \{c\}$.
• If $A = \{1,2,3\}$ and $B = \{4,5,6\}$, then $A \cap B = \emptyset$.

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Two sets $A$ and $B$ are disjoint if they have no elements in common, i.e., $A \cap B = \emptyset$. Examples:

• If $A = \{1,2,3\}$ and $B = \{4,5,6\}$, then $A$ and $B$ are disjoint.
• If $A = \{1,2,3\}$ and $B = \{3,4,5\}$, then $A$ and $B$ are not disjoint since $A \cap B = \{3\}$.