Intersection

The in­ter­sec­tion of two sets \(A\) and \(B\), de­noted \(A \cap B\), is the set of el­e­ments which are in both \(A\) and \(B\).

illustration of the output of an intersection

For­mally stated, where \(C = A \cap B\)

$$x \in C \leftrightarrow (x \in A \land x \in B)$$

That is, Iff \(x\) is in the in­ter­sec­tion \(C\), then \(x\) is in \(A\) and \(x\) is in \(B\).

For ex­am­ple,

  • \(\{1,2\} \cap \{2,3\} = \{2\}\)

  • \(\{1,2\} \cap \{8,9\} = \{\}\)

  • \(\{0,2,4,6\} \cap \{3,4,5,6\} = \{4,6\}\)