Intersection (set theory)

The intersection of two or more sets is the set of elements that are common to all of them. Thus, the intersection of the sets $\{1, 2, 3\}$ and $\{1, 3, 5\}$ is the set $\{1, 3\}$ .

Intersection is denoted by the symbol $\cap$ , so the preceding example could be written $\{1, 2, 3\} \cap \{1, 3, 5\} = \{1, 3\}$ . One can also use the symbol for intersection in the way one uses a capital sigma ( $\Sigma$ ) for sums, i.e. $\bigcap_{i = 1}^n A_i = A_1 \cap A_2 \cap \ldots \cap A_n$ is the intersection of the $n$ sets $A_1, A_2, \ldots, A_n$ .

Properties

For any sets $A, B$ , $A \cap B \subseteq A$ and $A \cap B \subseteq B$ . Thus $A \cap B = A$ if and only if $A \subseteq B$ .

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Intersection (set theory)

Properties

See also