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