From edb4827bf016a7ee9d5812a84bb29ebf48b8e22c Mon Sep 17 00:00:00 2001 From: Samuel Lubliner Date: Tue, 20 Feb 2024 20:34:37 +0000 Subject: [PATCH] Latest build deployed. --- equivalence.html | 94 +++++++++++++++++++++++++++--------- frontmatter.html | 2 +- index.html | 2 +- intro-relations.html | 2 +- lunr-pretext-search-index.js | 33 ++++++++----- partial-order.html | 20 ++++---- 6 files changed, 105 insertions(+), 48 deletions(-) diff --git a/equivalence.html b/equivalence.html index b82d558d..cff808b5 100644 --- a/equivalence.html +++ b/equivalence.html @@ -86,7 +86,19 @@ )} } }; - + @@ -221,41 +233,75 @@

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Section 5.4 Equivalence

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A relation is called an equivalence relation if the relation satisfies the following properties: reflexive symmetric and transitive.
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-
-Class of equivalence is defined by
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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.
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The class of equivalence for an element a in set A is defined by the set:
\begin{equation*} -[a] = \{x \in A | aRx\} +|a| = \{x \in A \; | \; xRa\} \end{equation*}
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The class of equivalence of a is the set of all elements in A that are related to a.
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This set comprises all elements in A that are related to a through the relation R, illustrating how elements are grouped into equivalence classes.
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Consider set A as defined by the scenario:
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\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*}
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Let R be the following relation on A:
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x R y if and only if x and y live in the same building.
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    -
  • -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.
    • +
    +
    In this context, let R be the relation on A described as follows:
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    +\begin{equation*} +\text{x R y iff x and y live in the same floor of the building.} +\end{equation*} +
    +
    +
    This relation demonstrates the properties of an equivalence relation:
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      +
    • +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.
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    -
    Class of equivalence
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    +
    For the class of equivalence, considering person a as an example:
    \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*}
    -
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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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diff --git a/frontmatter.html b/frontmatter.html index 466312c7..379562d1 100644 --- a/frontmatter.html +++ b/frontmatter.html @@ -225,7 +225,7 @@

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Zunaid Ahmed, Hellen Colman, Samuel Lubliner
Math Department
City Colleges of Chicago
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February 19, 2024
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February 20, 2024
PrevTopNext diff --git a/index.html b/index.html index 691860ca..dbe2a39c 100644 --- a/index.html +++ b/index.html @@ -2,7 +2,7 @@ - + diff --git a/intro-relations.html b/intro-relations.html index 00e81f69..16b90272 100644 --- a/intro-relations.html +++ b/intro-relations.html @@ -234,7 +234,7 @@

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Section 5.1 Introduction to Relations
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A relation \(R\) from set \(A\) into set \(B\) is defined as a subset of the Cartesian product \(A \times B\text{,}\) represented as:
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A relation \(R\) from set \(A\) into set \(B\) is defined as a subset of the Cartesian product \(A \times B\text{,}\) represented as:
\begin{equation*} R \subseteq A \times B diff --git a/lunr-pretext-search-index.js b/lunr-pretext-search-index.js index 2cf8b2af..a84341ad 100644 --- a/lunr-pretext-search-index.js +++ b/lunr-pretext-search-index.js @@ -241,7 +241,16 @@ var ptx_lunr_docs = [ "type": "Section", "number": "5.1", "title": "Introduction to Relations", - "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. " +}, +{ + "id": "intro-relations-2", + "level": "2", + "url": "intro-relations.html#intro-relations-2", + "type": "Paragraph (with a defined term)", + "number": "", + "title": "", + "body": "relation " }, { "id": "relations-on-a-set", @@ -313,7 +322,7 @@ var ptx_lunr_docs = [ "type": "Section", "number": "5.4", "title": "Equivalence", - "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. " }, { "id": "equivalence-2", @@ -331,12 +340,12 @@ var ptx_lunr_docs = [ "type": "Paragraph (with a defined term)", "number": "", "title": "", - "body": "Class of equivalence " + "body": "class of equivalence " }, { - "id": "equivalence-8", + "id": "equivalence-13", "level": "2", - "url": "equivalence.html#equivalence-8", + "url": "equivalence.html#equivalence-13", "type": "Paragraph (with a defined term)", "number": "", "title": "", @@ -349,30 +358,30 @@ var ptx_lunr_docs = [ "type": "Section", "number": "5.5", "title": "Partial Order", - "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 . " }, { - "id": "partial-order-3", + "id": "partial-order-3-1-1", "level": "2", - "url": "partial-order.html#partial-order-3", + "url": "partial-order.html#partial-order-3-1-1", "type": "Paragraph (with a defined term)", "number": "", "title": "", "body": "Reflexive " }, { - "id": "partial-order-4", + "id": "partial-order-3-2-1", "level": "2", - "url": "partial-order.html#partial-order-4", + "url": "partial-order.html#partial-order-3-2-1", "type": "Paragraph (with a defined term)", "number": "", "title": "", "body": "Antisymmetric " }, { - "id": "partial-order-5", + "id": "partial-order-3-3-1", "level": "2", - "url": "partial-order.html#partial-order-5", + "url": "partial-order.html#partial-order-3-3-1", "type": "Paragraph (with a defined term)", "number": "", "title": "", diff --git a/partial-order.html b/partial-order.html index 92c93344..41bc25b2 100644 --- a/partial-order.html +++ b/partial-order.html @@ -234,18 +234,20 @@

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Section 5.5 Partial Order
A Partial Order (PO) \(\prec\) satisfies the following properties:
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-Reflexive a R a for all a \(\in\) A
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    +
  • +Reflexive a R a for all a \(\in\) A
    • +
  • +Antisymmetric If a R b and b R a then a = b for all a, b \(\in\) A
    • +
  • +Transitive If a R b and b R c then a R c for all a, b, c \(\in\) A
    • +
-Antisymmetric If a R b and b R a then a = b for all a, b \(\in\) A
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-Transitive If a R b and b R c then a R c for all a, b, c \(\in\) A
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Example: Let \(B = \{0,1\}\) and define \(A\) as the power set of \(B\text{,}\) denoted \(A = \mathcal{P}(B)\text{.}\) Establish a relation \(R\) on \(A\) where \(XRY\) if and only if \(X \subseteq Y\text{.}\) This relation represents the idea of one set being a subset of another within the power set of \(B\text{.}\)
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To explore how elements relate within these examples, consider the element \(\{1\}\) in the context of the second example. The set \(\{1\}\) is not related to the empty set, denoted as \(\{\}\text{,}\) because \(\{1\}\) is not a subset of \(\{\}\text{.}\) Similarly, \(\{1\}\) does not relate to \(\{0\}\) because \(\{1\}\) is not a subset of \(\{0\}\text{.}\) However, \(\{1\}\) is related to \(\{0,1\}\) because \(\{1\}\) is indeed a subset of \(\{0,1\}\text{,}\) which we denote as \(\{1\} R \{0,1\}\text{.}\) +
To explore how elements relate within these examples, consider the element \(\{1\}\) in the context of the second example. The set \(\{1\}\) is not related to the empty set, denoted as \(\{\}\text{,}\) because \(\{1\}\) is not a subset of \(\{\}\text{.}\) Similarly, \(\{1\}\) does not relate to \(\{0\}\) because \(\{1\}\) is not a subset of \(\{0\}\text{.}\) However, \(\{1\}\) is related to \(\{0,1\}\) because \(\{1\}\) is indeed a subset of \(\{0,1\}\text{,}\) which we denote as \(\{1\} R \{0,1\}\text{.}\)
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-