# 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:

1. $$\overline{A\cap B} = (A\cap B)^{c}$$
2. $$B-A$$

### 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.

1. The First law: $\overline{\left(A_{1}\cup A_{2}\cup A_{3}\right)} = \overline{A_{1}} \cap \overline{A_{2}} \cap \overline{A_{3}}$

2. The Second Law: $\overline{\left(A_{1}\cap A_{2}\cap A_{3}\right)} = \overline{A_{1}} \cup \overline{A_{2}} \cup \overline{A_{3}}$