Question

Let A and B be subsets of the finite universal set U. Show that$\mathbf{|}\mathbf{¯}\mathit{A}\mathbf{\cap }\mathbf{¯}\mathit{B}\mathbf{|}\mathbf{=}\mathbf{\left|}\mathit{U}\mathbf{\right|}\mathbf{-}\mathbf{\left|}\mathit{A}\mathbf{\right|}\mathbf{-}\mathbf{\left|}\mathit{B}\mathbf{\right|}\mathbf{+}\mathbf{|}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{|}$

Short answer

$|¯\mathrm{A}\cap ¯\mathrm{B}|=\left|\mathrm{U}\right|-\left|\mathrm{A}\right|-\left|\mathrm{B}\right|+|\mathrm{A}\cap \mathrm{B}|$

Step-by-step solution

Step: 1

$$\begin{gathered} =|A\cup B{|}^{\epsilon }=|U|-(A\cup B) \\ =\left|U\right|-\left(\right|A|+|B|-|A\cap B\left|\right) \\ =\left|U\right|-\left|A\right|-\left|B\right|+|A\cap B| \end{gathered}$$

$|¯A\cap ¯B|=\left|U\right|-\left|A\right|-\left|B\right|+|A\cap B|$