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<section xml:id="sec-Set-Theory" xmlns:xi="http://www.w3.org/2001/XInclude"> <section xml:id="sec-Set-Theory" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Set Theory</title> <title>Set Theory</title>
<p> <p>
When we perform an experiment, there are many results that we might see. When we perform an experiment, there are many results that we might see.
We want to be able to quantify the likelihood of seeing certain results. We want to be able to quantify the likelihood of seeing certain results.
For this, we need to develop some mathematical terminology. For this, we need to develop some mathematical terminology.
</p> </p>
<definition xml:id="def-sample-space"> <definition xml:id="def-sample-space">
<statement> <statement>
<p> <p>
The <term>sample space</term>, often denoted <m>\Omega</m>, is the set of all possible results of an experiment. The <term>sample space</term>, often denoted <m>\Omega</m>, is the set
A single result is called an <term>outcome</term>, while a collection of results is called an <term>event</term>. of all possible results of an experiment.
</p> A single result is called an <term>outcome</term>, while a collection of
</statement> results is called an <term>event</term>.
</definition> </p>
</statement>
</definition>
<example xml:id="example-sample-space"> <example xml:id="example-sample-space">
<statement> <statement>
<p> <p>
An experiment consists of rolling a standard 6-sided die. An experiment consists of rolling a standard 6-sided die.
The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>. The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the result of the roll is even. One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the
</p> result of the roll is even.
</p>
<p> <p>
What would the sample space look like if we roll the die two times and recorded the results? What would the sample space look like if we roll the die two times and
</p> recorded the results?
</statement> </p>
</statement>
<answer> <answer>
<p> <p>
<md> <md>
<mrow> \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </mrow> <mrow> \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </mrow>
<mrow> \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </mrow> <mrow> \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
<mrow> \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </mrow> <mrow> \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </mrow>
<mrow> \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </mrow> <mrow> \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </mrow>
<mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow> <mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow>
<mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow> <mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
</md> </md>
Note that, for example, <m>(1, 2)</m> is a different outcome from <m>(2, 1)</m>. Note that, for example, <m>(1, 2)</m> is a different outcome from
</p> <m>(2, 1)</m>.
</answer> </p>
</example> </answer>
</example>
<definition xml:id="def-subset"> <definition xml:id="def-subset">
<statement> <statement>
<p> <p>
Let <m>A</m> be a set. Let <m>A</m> be a set.
The symbol <m>\in</m> means "is an element of", as in <m>a \in A</m>. The symbol <m>\in</m> means "is an element of", as in <m>a \in A</m>.
Given another set <m>B</m>, we say <m>A</m> is a <term>subset</term> of <m>B</m>, written <m>A\subset B</m>, to mean that every element of the set <m>A</m> is also an element of the set <m>B</m>. Given another set <m>B</m>, we say <m>A</m> is a <term>subset</term> of
</p> <m>B</m>, written <m>A\subset B</m>, to mean that every element of the
</statement> set <m>A</m> is also an element of the set <m>B</m>.
</definition> </p>
</statement>
</definition>
<p> <p>
The subset symbol includes the possibility that <m>A</m> and <m>B</m> are equal sets, i.e., that they contain precisely the same elements. The subset symbol includes the possibility that <m>A</m> and <m>B</m> are
</p> equal sets, i.e., that they contain precisely the same elements.
</p>
<definition xml:id="def-set-operations"> <definition xml:id="def-set-operations">
<statement> <statement>
<p> <p>
Consider sets <m>A</m> and <m>B</m>, each contained inside <m>\Omega</m>. Consider sets <m>A</m> and <m>B</m>, each contained inside
We can combine sets in a variety of ways: <dl> <m>\Omega</m>.
We can combine sets in a variety of ways:
<dl>
<li> <li>
<title>Union</title> <title>Union</title>
<p> <p>
The <term>union</term> of <m>A</m> and <m>B</m> is the set <m>A \cup B = \{x \mid x \in A \text{ or } x \in B\}</m>. The <term>union</term> of <m>A</m> and <m>B</m> is the set
<m>A \cup B = \{x \mid x \in A \text{ or } x \in B\}</m>.
</p> </p>
</li> </li>
@@ -75,7 +86,8 @@
<title>Intersection</title> <title>Intersection</title>
<p> <p>
The <term>intersection</term> of <m>A</m> and <m>B</m> is the set <m>A \cap B = \{x \mid x \in A \text{ and } x \in B\}</m>. The <term>intersection</term> of <m>A</m> and <m>B</m> is the set
<m>A \cap B = \{x \mid x \in A \text{ and } x \in B\}</m>.
</p> </p>
</li> </li>
@@ -83,7 +95,8 @@
<title>Difference</title> <title>Difference</title>
<p> <p>
The <term>set difference</term> <m>A-B</m> is the set <m>A - B = \{x \mid x \in A \text{ and } x \notin B\}</m>. The <term>set difference</term> <m>A-B</m> is the set
<m>A - B = \{x \mid x \in A \text{ and } x \notin B\}</m>.
</p> </p>
</li> </li>
@@ -91,7 +104,8 @@
<title>Complement</title> <title>Complement</title>
<p> <p>
The <term>complement</term> of <m>A</m> is the set <m>A^c = \{x \in \Omega \mid x \notin A\}</m>. The <term>complement</term> of <m>A</m> is the set
<m>A^c = \{x \in \Omega \mid x \notin A\}</m>.
</p> </p>
</li> </li>
@@ -99,27 +113,30 @@
<title>Empty Set</title> <title>Empty Set</title>
<p> <p>
The <term>empty set</term>, usually written <m>\emptyset</m> or <m>\{\}</m>, is the set which contains no elements. The <term>empty set</term>, usually written <m>\emptyset</m> or
<m>\{\}</m>, is the set which contains no elements.
</p> </p>
</li> </li>
</dl> </dl>
</p>
</statement>
</definition>
<p>
It's useful sometimes to draw pictures called <term>Venn diagrams</term>
representing sets:
</p>
<figure xml:id="fig-Venn-diagram">
<caption>Example Venn Diagram</caption>
<image width="50%">
<description>
<p>
Venn diagram showing sets <m>A, B, C</m> with the region representing
<m>(A\cup B\cup C) - (A \cap C)</m> shaded.
</p> </p>
</statement> </description>
</definition> <latex-image>
<p>
It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing sets:
</p>
<figure xml:id="fig-Venn-diagram">
<caption>Example Venn Diagram</caption>
<image width="50%">
<description>
<p>
Venn diagram showing sets <m>A, B, C</m> with the region representing <m>(A\cup B\cup C) - (A \cap C)</m> shaded.
</p>
</description>
<latex-image>
\begin{tikzpicture} \begin{tikzpicture}
\def\firstcircle{(90:1.75cm) circle (2.5cm)} \def\firstcircle{(90:1.75cm) circle (2.5cm)}
\def\secondcircle{(210:1.75cm) circle (2.5cm)} \def\secondcircle{(210:1.75cm) circle (2.5cm)}
@@ -137,35 +154,38 @@
\draw \thirdcircle node [text=black,below right] {$C$}; \draw \thirdcircle node [text=black,below right] {$C$};
\node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$}; \node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$};
\end{tikzpicture} \end{tikzpicture}
</latex-image> </latex-image>
</image> </image>
</figure> </figure>
<definition xml:id="def-disjoint"> <definition xml:id="def-disjoint">
<statement> <statement>
<p> <p>
Two sets <m>A</m> and <m>B</m> are <term>disjoint</term> if <m>A \cap B = \emptyset</m>. Two sets <m>A</m> and <m>B</m> are <term>disjoint</term> if
</p> <m>A \cap B = \emptyset</m>.
</statement> </p>
</definition> </statement>
</definition>
<definition xml:id="def-cardinality"> <definition xml:id="def-cardinality">
<statement> <statement>
<p> <p>
Given a finite set <m>A</m>, the <term>cardinality</term> of <m>A</m>, written <m>|A|</m>, is the number of elements in <m>A</m>. Given a finite set <m>A</m>, the <term>cardinality</term> of <m>A</m>,
</p> written <m>|A|</m>, is the number of elements in <m>A</m>.
</statement> </p>
</definition> </statement>
</definition>
<exercises xml:id="exercises-Set-Theory"> <exercises xml:id="exercises-Set-Theory">
<exercise> <exercise>
<introduction> <introduction>
<p> <p>
Consider the sets <m>A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</m>, <m>B = \{2, 4, 9, 10, 12, 14, 19\}</m>, and <m>C = \{9, 10, 11, 14, 16, 17, 20\}</m>, which are all subsets of <m>\Omega = \{1, 2, 3, \dotsc, 20\}</m>. Consider the sets <m>A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</m>,
<m>B = \{2, 4, 9, 10, 12, 14, 19\}</m>, and
<m>C = \{9, 10, 11, 14, 16, 17, 20\}</m>, which are all subsets of
<m>\Omega = \{1, 2, 3, \dotsc, 20\}</m>.
</p> </p>
</introduction> </introduction>
<task> <task>
<statement> <statement>
<p> <p>
@@ -178,24 +198,78 @@
<m>\{1, 2, 3, 4, 5, 6, 7, 8\}</m> <m>\{1, 2, 3, 4, 5, 6, 7, 8\}</m>
</p> </p>
</answer> </answer>
</task>
<solution>
<p>
The set <m>A - (B \cap C)</m> consists of elements in the set
<m>A</m> which are not in the overlap of <m>B</m> and <m>C</m>.
The overlap is <m>B \cap C = \{9, 10, 14\}</m>, and of these
elements, <m>9</m> and <m>10</m> are in <m>A</m>.
So <m>A - (B \cap C) = \{1, 2, 3, 4, 5, 6, 7, 8\}</m>.
</p>
</solution>
</task>
<task> <task>
<statement> <statement>
<p> <p>
Find <m>|A|</m>, <m>|B|</m>, <m>|C|</m>, <m>|A\cup B|</m>, <m>|A \cap B|</m>, <m>|B\cap C|</m>, <m>|A\cap C|</m>, and <m>|A\cup B\cup C|</m>. Find <m>|A|</m>, <m>|B|</m>, <m>|C|</m>, <m>|A\cup B|</m>,
Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets? <m>|A \cap B|</m>, <m>|B\cap C|</m>, <m>|A\cap C|</m>, and
<m>|A\cup B\cup C|</m>.
Is it true that the size of the union of sets is equal to the sum of
the sizes of the individual sets?
</p> </p>
</statement> </statement>
<answer> <answer>
<p> <p>
<m>|A| = 10</m>, <m>|B| = 7</m>, <m>|C| = 7</m>, <m>|A \cup B| = 13</m>, <m>|A \cap B| = 4</m>, <m>|B\cap C| = 3</m>, <m>|A\cap C| = 2</m>, <m>|A\cup B\cup C| = 17</m>. In particular, note that <m>|A\cup B| = 13 \neq 10 + 7 = |A| + |B|</m>, 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. <m>|A| = 10</m>, <m>|B| = 7</m>, <m>|C| = 7</m>,
<m>|A \cup B| = 13</m>, <m>|A \cap B| = 4</m>, <m>|B\cap C| = 3</m>,
<m>|A\cap C| = 2</m>, <m>|A\cup B\cup C| = 17</m>.
In particular, note that
<m>|A\cup B| = 13 \neq 10 + 7 = |A| + |B|</m>, 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.
</p> </p>
</answer> </answer>
</task>
<solution>
<p>
<m>|X|</m> counts the number of elements in a finite set <m>X</m>.
We can quickly count the elements in sets <m>A, B, C</m> to see that
<m>|A| = 10</m>, <m>|B| = 7</m>, and <m>|C| = 7</m>.
</p>
<p>
The union of two sets includes all elements from either set, so:
<md>
<mrow> A \cup B \amp = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 19\} </mrow>
<mrow> A \cup B \cup C \amp = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 20\} </mrow>
</md>
Counting the elements, we see <m>|A \cup B| = 13</m> and
<m>|A \cup B \cup C| = 17</m>.
</p>
<p>
The intersection of two sets is the overlap, consisting of elements
which show up in both sets simultaneously, so:
<md>
<mrow> A \cap B \amp = \{2, 4, 9, 10\} </mrow>
<mrow> B \cap C \amp = \{9, 10, 14\} </mrow>
<mrow> A \cap C \amp = \{9, 10\} </mrow>
</md>
Counting the elements, we see <m>|A\cap B| = 4</m>,
<m>|B \cap C| = 3</m>, and <m>|A\cap C| = 2</m>.
</p>
<p>
In particular, note that
<m>|A\cup B| = 13 \neq 10 + 7 = |A| + |B|</m>, 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.
</p>
</solution>
</task>
<task> <task>
<statement> <statement>
@@ -206,19 +280,33 @@
<answer> <answer>
<p> <p>
<m>A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}</m>, <m>(A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}</m>. <m>A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}</m>,
<m>(A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}</m>.
</p> </p>
</answer> </answer>
<solution>
<p>
<m>A^c</m> consists of elements of <m>\Omega</m> which are not in
<m>A</m>, thus
<m>A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}</m>.
Similarly, <m>(A\cup B)^c</m> consists of elements that are not in
<m>A</m> nor in <m>B</m>.
Therefore, <m>(A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}</m>.
</p>
</solution>
</task> </task>
</exercise> </exercise>
<exercise> <exercise>
<statement> <statement>
<p> <p>
Suppose we have a 6-sided die that's weighted to roll a 6 half of the time. Suppose we have a 6-sided die that's weighted to roll a 6 half of the
time.
We roll the die two times. We roll the die two times.
List the set of all possible results. List the set of all possible results.
[Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result <m>(4, 2)</m>---rolling a 4 and then a 2.] [Note: the result (2, 4)---rolling a 2 and then a 4---is different
from the result <m>(4, 2)</m>---rolling a 4 and then a 2.]
</p> </p>
</statement> </statement>
@@ -232,10 +320,25 @@
<mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow> <mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow>
<mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow> <mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
</md> </md>
Note that <m>\Omega</m> simply lists outcomes with no reference to the probabilities.
So the answer here is the same as in <xref ref="example-sample-space"/>.
</p> </p>
</answer> </answer>
<solution>
<p>
<md>
<mrow> \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </mrow>
<mrow> \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
<mrow> \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </mrow>
<mrow> \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </mrow>
<mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow>
<mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
</md>
Note that <m>\Omega</m> simply lists outcomes with no reference to the
probabilities.
So the answer here is the same as in
<xref ref="example-sample-space"/>.
</p>
</solution>
</exercise> </exercise>
<exercise> <exercise>
@@ -243,7 +346,9 @@
<p> <p>
Suppose we flip a coin two times. Suppose we flip a coin two times.
List the set of all possible results. 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? What about flipping three times?
Four times?
If we flip the coin 10 times, how many possible results will there be?
</p> </p>
</statement> </statement>
@@ -253,7 +358,11 @@
</p> </p>
<p> <p>
For three flips: <m>\Omega = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}</m>. For three flips:
<md>
<mrow> \Omega = \{ \amp HHH, HHT, HTH, THH, </mrow>
<mrow> \amp HTT, THT, TTH, TTT \}</mrow>
</md>
</p> </p>
<p> <p>
@@ -276,16 +385,23 @@
<exercise> <exercise>
<statement> <statement>
<p> <p>
If we roll a 6-sided die ten times, how many possible results will there be? If we roll a 6-sided die ten times, how many possible results will
there be?
</p> </p>
</statement> </statement>
<answer> <answer>
<p> <p>
Each additional roll will multiply the number of outcomes by 6. <m>|\Omega| = 6^{10}</m>.
So, with 10 rolls, we'll have <m>|\Omega| = 6^{10}.</m>
</p> </p>
</answer> </answer>
<solution>
<p>
Each additional roll will multiply the number of outcomes by 6.
So, with 10 rolls, we'll have <m>|\Omega| = 6^{10}</m>.
</p>
</solution>
</exercise> </exercise>
</exercises> </exercises>
</section> </section>