An engineer is asked to develop a glass break detector$/$discriminator for use with residential alarm systems. The detector should be able to distinguish between the breaking of a pane of a glass $($a window$)$ and a drinking glass. From analysis it has been determined that the sound of a shattering window pane contains most of its energy at frequencies centered about $5kHz$ whereas the sound of a shattering drinking glass contains most of its energy at frequencies centered about $8kHz.$ The spectra of the two shattering sounds overlap. The membership functions for the window pane and the glass are given as $x^4(x)$ and ${}_B(x),$ respectively. Illustrate the basic operations of union, intersection, complement, and difference for the following membership functions:

$x=0,1,$cdots,$10,=2\mathrm{.}5,{}_A=5,{}_B=8$

${}_A(x)=$exp$[\frac{-(x-{}_A{)}^2}{2{}^2}],{}_B(x)=$exp$[\frac{-(x-{}_B{)}^2}{2{}^2}]$