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)}
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</script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><link href="_static/pretext/css/pretext_search.css" rel="stylesheet" type="text/css">
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Expand Down Expand Up @@ -221,41 +233,75 @@ <h2 class="search-results-heading">Search Results: </h2>
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="equivalence"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space"> </span><span class="codenumber">5.4</span><span class="space"> </span><span class="title">Equivalence</span>
</h2>
<div class="para" id="equivalence-2">A relation is called an <dfn class="terminology">equivalence relation</dfn> if the relation satisfies the following properties: reflexive symmetric and transitive.</div>
<div class="para logical" id="equivalence-3">
<div class="para">
<dfn class="terminology">Class of equivalence</dfn> is defined by</div>
<div class="para" id="equivalence-2">A relation is called an <dfn class="terminology">equivalence relation</dfn> if it satisfies three key properties: reflexivity, symmetry, and transitivity. These properties ensure that elements can be grouped into classes of equivalence based on their mutual relations.</div>
<div class="para" id="equivalence-3">The <dfn class="terminology">class of equivalence</dfn> for an element a in set A is defined by the set:</div>
<div class="displaymath process-math">
\begin{equation*}
[a] = \{x \in A | aRx\}
|a| = \{x \in A \; | \; xRa\}
\end{equation*}
</div>
</div>
<div class="para" id="equivalence-4">The class of equivalence of a is the set of all elements in A that are related to a.</div>
<div class="para logical" id="equivalence-5"><div class="displaymath process-math">
<div class="para" id="equivalence-5">This set comprises all elements in A that are related to a through the relation R, illustrating how elements are grouped into equivalence classes.</div>
<div class="para" id="equivalence-6">Consider set A as defined by the scenario:</div>
<div class="para logical" id="equivalence-7"><div class="displaymath process-math">
\begin{equation*}
\text{Let A } = \{x | x \text{ is a person living in USA} \}
\text{Let A } = \{x \; | \; x \text{ is a person living in a given building} \}
\end{equation*}
</div></div>
<div class="para" id="equivalence-6">Let R be the following relation on A:</div>
<div class="para" id="equivalence-7">x R y if and only if x and y live in the same building.</div>
<div class="para logical" id="equivalence-8"><ul class="disc">
<li id="equivalence-8-1-1"><div class="para" id="equivalence-8-1-1-1">
<dfn class="terminology">Reflexive</dfn>: A person lives in the same building as himself. This is true for everybody living in USA.</div></li>
<li id="equivalence-8-1-2"><div class="para" id="equivalence-8-1-2-1">
<dfn class="terminology">Symmetric</dfn>: If person x lives in the same building as person y, then person y lives in the same building as person x.</div></li>
<li id="equivalence-8-1-3"><div class="para" id="equivalence-8-1-3-1">
<dfn class="terminology">Transitive</dfn>: If person x lives in the same building as person y and person y lives in the same building as person z, then person x lives in the same building as person z.</div></li>
<pre id="equivalence-8" class="ptx-sagecell sagecell-sage"><script type="text/x-sage"># Define the set of people and their respective floors using SageMath Set
A = Set([('Person1', 1), ('Person2', 2), ('Person3', 3),
('Person4', 1), ('Person5', 2), ('Person6', 3),
('Person7', 1), ('Person8', 2), ('Person9', 3), ('Person10', 1)])
</script></pre>
<div class="para" id="equivalence-9">In this context, let R be the relation on A described as follows:</div>
<div class="displaymath process-math">
\begin{equation*}
\text{x R y iff x and y live in the same floor of the building.}
\end{equation*}
</div>
<pre id="equivalence-11" class="ptx-sagecell sagecell-sage"><script type="text/x-sage">def R(x, y):
return x[1] == y[1] # x[1] and y[1] represent the floors of person x and y, respectively

def class_of_equivalence(person, A):
return Set([x for x in A if R(person, x)])

def print_class_of_equivalence(person, A):
equivalence_class = class_of_equivalence(person, A)
# Extracting just the names from the tuples for a more readable output
names_in_class = [x[0] for x in equivalence_class]
print(f"Class of Equivalence for {person[0]} (living on floor {person[1]}): {', '.join(names_in_class)}")

# Example: Class of equivalence for 'Person1'
person_a = ('Person1', 1)
print_class_of_equivalence(person_a, A)
</script></pre>
<div class="para" id="equivalence-12">This relation demonstrates the properties of an equivalence relation:</div>
<div class="para logical" id="equivalence-13"><ul class="disc">
<li id="equivalence-13-1-1"><div class="para" id="equivalence-13-1-1-1">
<dfn class="terminology">Reflexive</dfn>: A person lives in the same floor as themselves.</div></li>
<li id="equivalence-13-1-2"><div class="para" id="equivalence-13-1-2-1">
<dfn class="terminology">Symmetric</dfn>: If person x lives in the same floor as person y, then person y lives in the same floor as person x.</div></li>
<li id="equivalence-13-1-3"><div class="para" id="equivalence-13-1-3-1">
<dfn class="terminology">Transitive</dfn>: If person x lives in the same floor as person y and person y lives in the same floor as person z, then person x lives in the same floor as person z.</div></li>
</ul></div>
<div class="para logical" id="equivalence-9">
<div class="para">Class of equivalence</div>
<pre id="equivalence-14" class="ptx-sagecell sagecell-sage"><script type="text/x-sage"># Reflexivity Test
reflexive_test = all(R(person, person) for person in A)
print("Reflexive:", reflexive_test)

# Symmetry Test
symmetry_test = all(R(x, y) == R(y, x) for x in A for y in A)
print("Symmetric:", symmetry_test)

# Transitivity Test
transitivity_test = all(not R(x, y) or not R(y, z) or R(x, z) for x in A for y in A for z in A)
print("Transitive:", transitivity_test)
</script></pre>
<div class="para" id="equivalence-15">For the class of equivalence, considering person a as an example:</div>
<div class="displaymath process-math">
\begin{equation*}
\text{person } a = \{ x \in A | x R a \} =
\text{all people living in the same building as person } a
| \text{person a} | = \{ x \in A \; | \; x R a \} = \text{all people living on the same floor as person a}
\end{equation*}
</div>
</div></section></div>
<div class="para" id="equivalence-17">This definition shows that the class of equivalence for person a includes all individuals residing on the same floor as a. The relation "living on the same floor as" groups the building’s residents into sets, with each set corresponding to a floor, forming an equivalence class.</div></section></div>
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2 changes: 1 addition & 1 deletion frontmatter.html
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Expand Up @@ -225,7 +225,7 @@ <h2 class="search-results-heading">Search Results: </h2>
<div class="author-name">Zunaid Ahmed, Hellen Colman, Samuel Lubliner</div>
<div class="author-info">Math Department<br>City Colleges of Chicago</div>
</div>
<div class="date">February 19, 2024</div>
<div class="date">February 20, 2024</div>
<nav class="summary-links"><ul><li><a href="frontmatter-2.html" class="internal"><span class="title">Colophon</span></a></li></ul></nav></section></div>
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<html>
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<span class="type">Section</span><span class="space"> </span><span class="codenumber">5.1</span><span class="space"> </span><span class="title">Introduction to Relations</span>
</h2>
<div class="para logical" id="intro-relations-2">
<div class="para">A relation <span class="process-math">\(R\)</span> from set <span class="process-math">\(A\)</span> into set <span class="process-math">\(B\)</span> is defined as a subset of the Cartesian product <span class="process-math">\(A \times B\text{,}\)</span> represented as:</div>
<div class="para">A <dfn class="terminology">relation</dfn> <span class="process-math">\(R\)</span> from set <span class="process-math">\(A\)</span> into set <span class="process-math">\(B\)</span> is defined as a subset of the Cartesian product <span class="process-math">\(A \times B\text{,}\)</span> represented as:</div>
<div class="displaymath process-math">
\begin{equation*}
R \subseteq A \times B
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33 changes: 21 additions & 12 deletions lunr-pretext-search-index.js
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Expand Up @@ -241,7 +241,16 @@ var ptx_lunr_docs = [
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"body": " Introduction to Relations A relation from set into set is defined as a subset of the Cartesian product , represented as: The Cartesian product consists of all possible ordered pairs , where and . Each pair combines an element from set with an element from set . Let's define two sets, Pants and Shirts, as examples: The Cartesian product of Pants and Shirts includes all possible combinations of pants with shirts. To explore how we can match pants with shirts based on their style, we define a relation as a subset of this Cartesian product, where each pair matches by a certain criterion, such as style. First, let's define a set of Styles, and then create relations for Pants and Shirts based on these styles. We can visualize all possible combinations of pants and shirts, taking their styles into account. With these combinations, let's define a specific relation from Pants to Shirts based on matching styles. This relation is a subset of the Cartesian product PantStyles ShirtStyles, including only those pairs where the pants and shirts have the same style. This is a second example of a relation, on another relation. "
"body": " Introduction to Relations A relation from set into set is defined as a subset of the Cartesian product , represented as: The Cartesian product consists of all possible ordered pairs , where and . Each pair combines an element from set with an element from set . Let's define two sets, Pants and Shirts, as examples: The Cartesian product of Pants and Shirts includes all possible combinations of pants with shirts. To explore how we can match pants with shirts based on their style, we define a relation as a subset of this Cartesian product, where each pair matches by a certain criterion, such as style. First, let's define a set of Styles, and then create relations for Pants and Shirts based on these styles. We can visualize all possible combinations of pants and shirts, taking their styles into account. With these combinations, let's define a specific relation from Pants to Shirts based on matching styles. This relation is a subset of the Cartesian product PantStyles ShirtStyles, including only those pairs where the pants and shirts have the same style. This is a second example of a relation, on another relation. "
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Expand Down Expand Up @@ -313,7 +322,7 @@ var ptx_lunr_docs = [
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"body": " Equivalence A relation is called an equivalence relation if the relation satisfies the following properties: reflexive symmetric and transitive. Class of equivalence is defined by The class of equivalence of a is the set of all elements in A that are related to a. Let R be the following relation on A: x R y if and only if x and y live in the same building. Reflexive : A person lives in the same building as himself. This is true for everybody living in USA. Symmetric : If person x lives in the same building as person y, then person y lives in the same building as person x. Transitive : If person x lives in the same building as person y and person y lives in the same building as person z, then person x lives in the same building as person z. Class of equivalence "
"body": " Equivalence A relation is called an equivalence relation if it satisfies three key properties: reflexivity, symmetry, and transitivity. These properties ensure that elements can be grouped into classes of equivalence based on their mutual relations. The class of equivalence for an element a in set A is defined by the set: This set comprises all elements in A that are related to a through the relation R, illustrating how elements are grouped into equivalence classes. Consider set A as defined by the scenario: In this context, let R be the relation on A described as follows: This relation demonstrates the properties of an equivalence relation: Reflexive : A person lives in the same floor as themselves. Symmetric : If person x lives in the same floor as person y, then person y lives in the same floor as person x. Transitive : If person x lives in the same floor as person y and person y lives in the same floor as person z, then person x lives in the same floor as person z. For the class of equivalence, considering person a as an example: This definition shows that the class of equivalence for person a includes all individuals residing on the same floor as a. The relation \"living on the same floor as\" groups the building's residents into sets, with each set corresponding to a floor, forming an equivalence class. "
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Expand All @@ -331,12 +340,12 @@ var ptx_lunr_docs = [
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Expand All @@ -349,30 +358,30 @@ var ptx_lunr_docs = [
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"body": " Partial Order A Partial Order (PO) satisfies the following properties: Reflexive a R a for all a A Antisymmetric If a R b and b R a then a = b for all a, b A Transitive If a R b and b R c then a R c for all a, b, c A Example: Let and define as the power set of , denoted . Establish a relation on where if and only if . This relation represents the idea of one set being a subset of another within the power set of . To explore how elements relate within these examples, consider the element in the context of the second example. The set is not related to the empty set, denoted as , because is not a subset of . Similarly, does not relate to because is not a subset of . However, is related to because is indeed a subset of , which we denote as . "
"body": " Partial Order A Partial Order (PO) satisfies the following properties: Reflexive a R a for all a A Antisymmetric If a R b and b R a then a = b for all a, b A Transitive If a R b and b R c then a R c for all a, b, c A Example: Let and define as the power set of , denoted . Establish a relation on where if and only if . This relation represents the idea of one set being a subset of another within the power set of . To explore how elements relate within these examples, consider the element in the context of the second example. The set is not related to the empty set, denoted as , because is not a subset of . Similarly, does not relate to because is not a subset of . However, is related to because is indeed a subset of , which we denote as . "
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