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andyeisenberg
@@ -121,11 +121,11 @@
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<!-- Whether to knowl a particular elements is set here -->
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<!-- Lots of elements have this possibility; see the guide -->
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<!--<knowl
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<knowl
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theorem="no"
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proof="yes"
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definition="no"
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example="yes"
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example="no"
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example-solution="yes"
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project="no"
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task="no"
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@@ -140,7 +140,7 @@
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exercise-divisional="no"
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exercise-worksheet="no"
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exercise-readingquestion="no"
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/>-->
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/>
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<!-- Specify the theme for the html by giving names to -->
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<!-- override defaults. Ex:<css theme="denver"/> -->
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+138
-2
@@ -12,9 +12,121 @@
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<section xml:id="sec-Set-Theory" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Set Theory</title>
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<definition xml:id="def-sample-space">
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<statement>
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<p>
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Text of section.
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The <term>sample space</term>, often denoted <m>\Omega</m>, is the set of all possible results of an experiment.
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A single result is called an <term>outcome</term>, while a collection of results is called an <term>event</term>.
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</p>
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</statement>
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</definition>
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<example>
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<statement>
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<p>
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An experiment consists of rolling a standard 6-sided die.
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The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
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One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the result of the roll is even.
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</p>
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<p>
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What would the sample space look like if we roll the die two times and recorded the results?
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</p>
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</statement>
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<answer>
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<p>
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<md>
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<mrow> \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </mrow>
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<mrow> \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
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<mrow> \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </mrow>
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<mrow> \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </mrow>
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<mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow>
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<mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
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</md>
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</p>
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</answer>
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</example>
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<definition xml:id="def-subset">
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<statement>
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<p>
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Let <m>A</m> be a set.
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The symbol <m>\in</m> means "is an element of", as in <m>a \in A</m>.
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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>.
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</p>
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</statement>
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</definition>
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<definition xml:id="def-set-operations">
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<statement>
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<p>
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Consider sets <m>A</m> and <m>B</m>, each contained inside <m>\Omega</m>.
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We can combine sets in a variety of ways:
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</p>
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<dl>
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<li>
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<title>Union</title>
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<p>
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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>.
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</p>
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</li>
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<li>
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<title>Intersection</title>
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<p>
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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>.
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</p>
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</li>
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<li>
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<title>Difference</title>
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<p>
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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>.
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</p>
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</li>
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<li>
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<title>Complement</title>
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<p>
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The <term>complement</term> of <m>A</m> is the set <m>A^c = \{x \in \Omega \mid x \notin A\}</m>.
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</p>
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</li>
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<li>
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<title>Empty Set</title>
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<p>
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The <term>empty set</term>, usually written <m>\emptyset</m> or <m>\{\}</m>, is the set which contains no elements.
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</p>
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</li>
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</dl>
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</statement>
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</definition>
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<p>
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It's useful sometimes to draw pictures representing sets...
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</p>
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<!-- TODO learn figures -->
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<definition xml:id="def-disjoint">
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<statement>
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<p>
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Two sets <m>A</m> and <m>B</m> are <term>disjoint</term> if <m>A \cap B = \emptyset</m>.
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</p>
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</statement>
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</definition>
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<definition xml:id="def-cardinality">
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<statement>
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<p>
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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>.
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</p>
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</statement>
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</definition>
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<exercises xml:id="exercises-Set-Theory">
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<exercise>
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@@ -104,10 +216,34 @@
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<section xml:id="sec-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Definition of Probability</title>
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<definition xml:id="def-probability-distribution">
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<statement>
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<p>
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Text of section.
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A <term>probability distribution</term> on a sample space <m>\Omega</m> assigns probabilities to every event, satisfying the following conditions:
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</p>
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<ol>
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<li>
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<p>
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<m>\Pr(\Omega) = 1</m>.
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</p>
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</li>
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<li>
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<p>
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<m>0 \leq \Pr(A) \leq 1</m> for any event <m>A</m>.
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</p>
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</li>
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<li>
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<p>
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If <m>A \cap B = \emptyset</m>, then <m>\Pr(A\cup B) = \Pr(A) + \Pr(B)</m>.
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</p>
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</li>
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</ol>
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</statement>
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</definition>
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<exercises xml:id="exercises-Probability">
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<exercise>
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<statement>
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@@ -7,21 +7,21 @@
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<bibinfo>
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<author>
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<personname>You</personname>
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<department>Your department</department>
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<institution>Your institution</institution>
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<personname>Andy Eisenberg</personname>
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<department>Mathematics Department</department>
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<institution>Temple University</institution>
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</author>
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<date>
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<today />
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</date>
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<website>
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<!-- <website>
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<url href="https://pretextbook.org">My Website</url>
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</website>
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</website> -->
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<copyright>
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<year>2020<ndash />2024</year>
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<holder>You</holder>
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<year>2025</year>
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<holder>Andy Eisenberg</holder>
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<shortlicense> This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit <url href="http://creativecommons.org/licenses/by-sa/4.0/" visual="creativecommons.org/licenses/by-sa/4.0"> CreativeCommons.org</url>
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</shortlicense>
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</copyright>
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