Question

Use Venn diagrams

$\left(a\right)$to simplify the expressions $(E\cup F)(E\cup F^c)$;

$\left(b\right)$to prove DeMorgan’s laws for events$E$and$F$. [That is, prove${(E\cup F)}^c=E^c{F}^c$, and ${\left(EF\right)}^c=E^c\cup F^c]$

Short answer

By using the commutative, distributive, and associative properties of$(\cup ,\cap )$

$a)E$

$b)EF$

$c)$$F\cup EG$

Step-by-step solution

Given Information.

$\left(a\right)expressions(E\cup F)(E\cup Fc);$

$\left(b\right)(E\cup F)c=EcFc,and\left(EF\right)c=Ec\cup Fc.$

Part (a) Explanation.

a) Simplification of $(E\cup F)\left(E\cup F^c\right)$

$(E\cup F)\left(E\cup F^c\right)$$=(E\cup F)E\cup (E\cup F)F^c$

$=E\cup (E\cup F)F^c$

$=E\cup \left[E\left(F^c\right)\cup F\left(F^c\right)\right]$

$=E\cup E\left(F^c\right)\cup F\left(F^c\right)$

$=E\cup E\left(F^c\right)\cup \varnothing$

$=E\cup E\left(F^c\right)$

$=E$

part (b) Explanation.

b) Simplification of $(E\cup F)\left(E^c\cup F\right)\left(E\cup F^c\right)$

$(E\cup F)\left(E^c\cup F\right)\left(E\cup F^c\right)$$=(E\cup F)\left(E\cup F^c\right)\left(E^c\cup F\right)$

$=\left[(E\cup F)\left(E\cup F^c\right)\right]\left(E^c\cup F\right)$

$=E\left(E^c\cup F\right)$

$=E\left(E^c\right)\cup EF$

$=\varnothing \cup EF$

$=EF$

part (c) Explanation.

c) Simplification of $(E\cup F)(F\cup G)$

$(E\cup F)(F\cup G)=(E\cup F)F\cup (E\cup F)G$

$=F\cup (E\cup F)G$

$=F\cup EG\cup FG$

$=F\cup FG\cup EG$

$=F\cup EG$