Q. 2.3 - A First Course in Probability

Question

$F = FE \cup F{E}^c$and$E \cup F = E \cup E^{c}F$

Step-by-Step Solution

Answer

Use associative and distributive laws.

$F=FE\cup F{E}^c\text{ and }E\cup F=E\cup E^{c}F$

Step 1 Given Information.

$F=FE\cup F{E}^c\text{ and }E\cup F=E\cup E^{c}F$

Step 2 Explanation.

$\begin{array}{rrr} F=FE\cup F{E}^c& \text{ and }E\cup F=E\cup E^{c}F & \\ F& =F\cap S & A\subseteq B\Rightarrow A\cap B=A\text{ (from def. of intersection) }\\ & =F\cap \left(E\cup E^c\right) & S=E\cup E^c\text{ (from def. of complement) }\\ & =FE\cup F{E}^c & \text{ distributive law }\end{array}$

This proves:

$F=FE\cup F{E}^c$

$E\cup F=E\cup \left(FE\cup F{E}^c\right)$ by previously proven identity

         $=(E\cup FE)\cup F{E}^c$          associative law

          $=E\cup F{E}^c A\subseteq B\Rightarrow A\cup B=B$ (from def. of the union)

This proves:

$E\cup F=E\cup F{E}^c$