Intersection

From CS2800 wiki


Definition: Intersection
If [math]A [/math] and [math]B [/math] are sets, then the intersection of [math]A [/math] and [math]B [/math] (written [math]A \href{/cs2800/wiki/index.php/%5Ccap}{\cap} B [/math]) is given by [math]A \href{/cs2800/wiki/index.php/%5Ccap}{\cap} B \href{/cs2800/wiki/index.php/Definition}{:=} \{x \href{/cs2800/wiki/index.php/%5Cmid}{\mid} x \href{/cs2800/wiki/index.php/%5Cin}{\in} A \href{/cs2800/wiki/index.php/And}{\text{ and }} x \href{/cs2800/wiki/index.php/%5Cin}{\in} B\} [/math].

This means that if you know [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A \href{/cs2800/wiki/index.php/%5Ccap}{\cap} B [/math], you can conclude 'both' that [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] 'and' [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} B [/math], and similarly you must prove both [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] and [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} B [/math] to prove [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A \href{/cs2800/wiki/index.php/%5Ccap}{\cap} B [/math].

In this Venn diagram, the intersection of [math]A [/math] and [math]B [/math] is shaded:

Venn-intersection.svg