Union, intersection and complement

\(A\cup B\) includes elements in either set, including both. \(A\cap B\) includes only elements common to both. \(A^{\prime}\) consists of elements of \(\xi\) outside \(A\). Complements always depend on the chosen universal set.

Example

Let \(\xi=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6\}\) and \(B=\{3,6\}\). Place each element in exactly one Venn region.

\(A\cup B=\{2,3,4,6\}\), \(A\cap B=\{6\}\) and \(A^{\prime}=\{1,3,5,7,8\}\).

To shade \(A\cap B^{\prime}\), select the part inside \(A\) and outside \(B\): the region containing \(2,4\). To shade \((A\cup B)^{\prime}\), select the region outside both circles but inside the rectangle. A complement does not include anything outside \(\xi\).

If sets have no common elements, they are disjoint and \(A\cap B=\varnothing\). This is different from one set being contained in the other.