Definition 1.1.1.
The sample space, often denoted \(\Omega\text{,}\) is the set of all possible results of an experiment. A single result is called an outcome, while a collection of results is called an event.
Section 1.1 Set Theory
Example 1.1.2.
An experiment consists of rolling a standard 6-sided die. The sample space is \(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\) One possible event is \(A = \{2, 4, 6\}\text{,}\) i.e., the event that the result of the roll is even.
What would the sample space look like if we roll the die two times and recorded the results?
Answer.
\begin{align*}
\Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}
\end{align*}
Definition 1.1.3.
Let \(A\) be a set. The symbol \(\in\) means "is an element of", as in \(a \in A\text{.}\) Given another set \(B\text{,}\) we say \(A\) is a subset of \(B\text{,}\) written \(A\subset B\text{,}\) to mean that every element of the set \(A\) is also an element of the set \(B\text{.}\)
Definition 1.1.4.
Consider sets \(A\) and \(B\text{,}\) each contained inside \(\Omega\text{.}\) We can combine sets in a variety of ways:
- Union
- The union of \(A\) and \(B\) is the set \(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\text{.}\)
- Intersection
- The intersection of \(A\) and \(B\) is the set \(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\text{.}\)
- Difference
- The set difference \(A-B\) is the set \(A - B = \{x \mid x \in A \text{ and } x \notin B\}\text{.}\)
- Complement
- Empty Set
- The empty set, usually written \(\emptyset\) or \(\{\}\text{,}\) is the set which contains no elements.
It’s useful sometimes to draw pictures representing sets...
Definition 1.1.5.
Definition 1.1.6.
Given a finite set \(A\text{,}\) the cardinality of \(A\text{,}\) written \(|A|\text{,}\) is the number of elements in \(A\text{.}\)
Exercises Exercises
1.
Consider the sets \(A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\text{,}\) \(B = \{2, 4, 9, 10, 12, 14, 19\}\text{,}\) and \(C = \{9, 10, 11, 14, 16, 17, 20\}\text{,}\) which are all subsets of \(\Omega = \{1, 2, 3, \dotsc, 20\}\text{.}\)
(a)
Find \(A - (B \cap C)\text{.}\)
(b)
Find \(|A|\text{,}\) \(|B|\text{,}\) \(|C|\text{,}\) \(|A\cup B|\text{,}\) \(|A \cap B|\text{,}\) \(|B\cap C|\text{,}\) \(|A\cap C|\text{,}\) and \(|A\cup B\cup C|\text{.}\) Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets?
Answer.
\(|A| = 10\text{,}\) \(|B| = 7\text{,}\) \(|C| = 7\text{,}\) \(|A \cup B| = 13\text{,}\) \(|A \cap B| = 4\text{,}\) \(|B\cap C| = 3\text{,}\) \(|A\cap C| = 2\text{,}\) \(|A\cup B\cup C| = 17\text{.}\) In particular, note that \(|A\cup B| = 13 \neq 10 + 7 = |A| + |B|\text{,}\) so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets.
(c)
2.
Suppose we roll a 6-sided die two times. List the set of all possible results. [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result \((4, 2)\)---rolling a 4 and then a 2.]
3.
Suppose we flip a coin two times. List the set of all possible results. What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be?
4.
If we roll a 6-sided die ten times, how many possible results will there be?
