# 10 📝 Exercise Solutions

## 10.2 Chapter 2

### 10.2.1 Exercise 2.1

*Consider our simple card game example. Build Venn Diagrams of the events “drawing a King” and “drawing a Queen.” Do they have element in common? What does that say about these events?*

### 10.2.2 Exercise 2.2

*Coming back to our example with the card game, build Venn diagrams for the events “drawing a King” and “drawing a Heart.” Do they have element in common? What does that say about these events?*

### 10.2.3 Exercise 2.3

*Let \(A\) and \(B\) be two non-exclusive Events in the Sample Space \(S\). Use Venn diagrams to represent:*

- \(\overline{A\cap B} = (A\cap B)^{c}\)
- \(B-A\)

### 10.2.4 Exercise 2.4

*Any segment of the Real line is an Uncountable set. Consider the segment ranging from (but excluding) 0 to 1 the Sample Space*
\[S = \{x : 0 <x <1\}\]
*and the sets*
\[\begin{align*}
A &= \{x \in S : 0.4<x<0.8\} \text{and}\\
B &= \{x \in S : 0.6<x<1\}
\end{align*}\]

*Using these definitions, compute:*

- \(A^c\)
- \(B^c\)
- \(B^c \cup A\)
- \(B^c \cup A^c\)
- \(A \cup B\)
- \(A \cap B\)
- \(B \cup A^c\)
- \(A^c \cup A\)

### 10.2.5 Exercise 2.5

*For each coin we have \(H\) for Head and \(T\) for Tail. Remember that the sample space contains the following four points:*

\[S = \{ (HH),(HT),(TH),(TT) \}.\]

*And consider the events:*

*\(A= H\) is obtained at least once = \(\Big\{ (HH),(HT),(TH) \Big\}\)**\(B=\) the second toss yields \(T\) = \(\Big\{ (HT),(TT) \Big\}\)*

*Using these definitions of \(A\) and \(B\) compute:*

- \(A^c\)
- \(B^c\)
- \(B^c \cup A\)
- \(A \cup B\)
- \(A \cap B\)
- \(B \cup A^c\)

### 10.2.6 Exercise 2.6

*Use Venn Diagrams to illustrate these relationships*

- \(A \cap S = A\);
- \(A \cup S = S\);
- \(A \cap \varnothing = \varnothing\);
- \(A \cup \varnothing = A\);
- \(A \cap A^c = \varnothing\);
- \(A \cup A^c = S\);
- \(A \cap A = A\);
- \(A \cup A = A\);

### 10.2.7 Exercise 2.7

*Use Venn Diagrams to show the First DeMorgan Law*

*Let \(A\) and \(B\) be two sets in \(S\), then*
\[\begin{eqnarray}
(A\cap B)^{c} =A^c \cup B^c,
\end{eqnarray}\]

### 10.2.8 Exercise 2.8

*Use Venn Diagrams to show the Second DeMorgan Law*

Let \(A\) and \(B\) be two sets in \(S\). Then:

\[\begin{eqnarray} (A\cup B)^{c} =A^c \cap B^c, \end{eqnarray}\]

### 10.2.9 Exercise 2.8

*Use Venn diagrams to show DeMorgan’s Laws for three sets.*

The

**First law**: \[\overline{\left(A_{1}\cup A_{2}\cup A_{3}\right)} = \overline{A_{1}} \cap \overline{A_{2}} \cap \overline{A_{3}}\]The

**Second Law**: \[\overline{\left(A_{1}\cap A_{2}\cap A_{3}\right)} = \overline{A_{1}} \cup \overline{A_{2}} \cup \overline{A_{3}}\]