Chapter 2: Q. 2.7 (page 52)

Use Venn diagrams
$(a)$ to simplify the expressions $(E \cup F) (E \cup F^{c})$ ;
$(b)$ to prove DeMorgan’s laws for events $E$ and $F$ . [That is, prove ${(E \cup F)}^{c} = E^{c} F^{c}$ , and ${(E F)}^{c} = E^{c} \cup F^{c}]$

Short Answer

Expert verified

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

$a) E$

$b) E F$

$c)$ $F \cup E G$

Step by step solution

Given Information.

$(a) e x p r e s s i o n s (E \cup F) (E \cup F c);$

$(b) (E \cup F) c = E c F c, a n d (E F) c = E c \cup F c .$

Part (a) Explanation.

a) Simplification of $(E \cup F) (E \cup F^{c})$

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

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

$= E \cup [E (F^{c}) \cup F (F^{c})]$

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

$= E \cup E (F^{c}) \cup \emptyset$

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

$= E$

part (b) Explanation.

b) Simplification of $(E \cup F) (E^{c} \cup F) (E \cup F^{c})$

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

$= [(E \cup F) (E \cup F^{c})] (E^{c} \cup F)$

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

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

$= \emptyset \cup E F$

$= E F$

part (c) Explanation.

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

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

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

$= F \cup E G \cup F G$

$= F \cup F G \cup E G$

$= F \cup E G$

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Use Venn diagrams
$(a)$ to simplify the expressions $(E \cup F) (E \cup F^{c})$ ;
$(b)$ to prove DeMorgan’s laws for events $E$ and $F$ . [That is, prove ${(E \cup F)}^{c} = E^{c} F^{c}$ , and ${(E F)}^{c} = E^{c} \cup F^{c}]$

Short Answer

Step by step solution

Given Information.

Part (a) Explanation.

part (b) Explanation.

part (c) Explanation.

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Use Venn diagrams(a)to simplify the expressions (E∪F)(E∪Fc);(b)to prove DeMorgan’s laws for eventsEandF. [That is, prove(E∪F)c=EcFc, and (EF)c=Ec∪Fc]

Short Answer

Step by step solution

Given Information.

Part (a) Explanation.

part (b) Explanation.

part (c) Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

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Use Venn diagrams
$(a)$ to simplify the expressions $(E \cup F) (E \cup F^{c})$ ;
$(b)$ to prove DeMorgan’s laws for events $E$ and $F$ . [That is, prove ${(E \cup F)}^{c} = E^{c} F^{c}$ , and ${(E F)}^{c} = E^{c} \cup F^{c}]$