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@ -570,8 +570,8 @@ END%%
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%%ANKI
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Basic
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Is $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a relation?
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Back: Yes.
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*Why* isn't $\{\langle x, y \rangle, \langle x, y, z \rangle\}$ a relation?
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Back: N/A. It is.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718329620187-->
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END%%
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@ -654,27 +654,20 @@ END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a\}$?
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Back: Yes.
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*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ reflexive on $\{a\}$?
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Back: N/A. It is.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720967429812-->
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END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a, b\}$?
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Back: No.
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*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ reflexive on $\{a, b\}$?
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Back: Because $\langle b, b \rangle \not\in R$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720967429817-->
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END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a, b\}$?
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Back: No.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, *why* isn't $R$ reflexive on $\{a, b\}$?
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@ -736,23 +729,23 @@ END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive on $\{a\}$?
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Back: No.
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*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ irreflexive on $\{a\}$?
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Back: Because $\langle a, a \rangle \in R$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1721870888395-->
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END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive on $\{b\}$?
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Back: Yes.
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*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ irreflexive on $\{b\}$?
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Back: N/A. It is.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1721870888400-->
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END%%
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%%ANKI
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Basic
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|
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, *why* isn't $R$ irreflexive on $\{a, b\}$?
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|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ irreflexive on $\{a, b\}$?
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Back: Because $\langle a, a \rangle \in R$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1721870888406-->
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@ -819,8 +812,8 @@ END%%
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%%ANKI
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Basic
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Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ symmetric?
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Back: No.
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*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle\}$ symmetric?
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Back: Because $aRb$ and $bRc$ but $\neg bRa$ and $\neg cRb$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720967429832-->
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END%%
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@ -863,28 +856,20 @@ END%%
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%%ANKI
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Basic
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Is $R = \{\langle a, a \rangle, \langle a, b \rangle, \langle b, a \rangle, \langle b, c \rangle\}$ antisymmetric?
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Back: No.
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*Why* isn't $R = \{\langle a, a \rangle, \langle a, b \rangle, \langle b, a \rangle, \langle b, c \rangle\}$ antisymmetric?
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Back: Because $aRb$ and $bRa$.
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|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
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<!--ID: 1721909725690-->
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END%%
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%%ANKI
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Basic
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|
Is $R = \{\langle a, a \rangle, \langle b, b \rangle, \langle b, c \rangle\}$ antisymmetric?
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Back: Yes.
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|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle, \langle b, c \rangle\}$ antisymmetric?
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Back: N/A. It is.
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|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
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<!--ID: 1721909725693-->
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END%%
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%%ANKI
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Basic
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|
*Why* isn't $R = \{\langle a, a \rangle, \langle a, b \rangle, \langle b, a \rangle, \langle b, c \rangle\}$ antisymmetric?
|
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Back: Because $aRb$ and $bRa$.
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|
Reference: “Antisymmetric Relation,” in _Wikipedia_, January 24, 2024, [https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation](https://en.wikipedia.org/w/index.php?title=Antisymmetric_relation&oldid=1198625107).
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<!--ID: 1721909725696-->
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END%%
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%%ANKI
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Basic
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|
Can a nonempty relation be both reflexive and antisymmetric on the same set?
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|
@ -970,7 +955,7 @@ END%%
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%%ANKI
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Basic
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|
What distinguishes the antecedent of antisymmetry's and asymmetric's FOL definition?
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|
What distinguishes the antecedent of antisymmetry's and asymmetry's FOL definition?
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|
Back: The former only considers *distinct* pairs of elements.
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|
Reference: “Asymmetric Relation,” in _Wikipedia_, February 21, 2024, [https://en.wikipedia.org/w/index.php?title=Asymmetric_relation](https://en.wikipedia.org/w/index.php?title=Asymmetric_relation&oldid=1209290822).
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<!--ID: 1721910949017-->
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@ -1122,20 +1107,12 @@ END%%
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%%ANKI
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Basic
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|
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ transitive?
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|
Back: No.
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|
*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle\}$ transitive?
|
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|
Back: Because $\langle a, c \rangle \not\in R$.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720967429846-->
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END%%
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%%ANKI
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Basic
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|
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, what additional member(s) must be added to make $R$ transitive?
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Back: Just $\langle a, c \rangle$.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720967429850-->
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END%%
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%%ANKI
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Basic
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|
Which of symmetric relations and transitive relations is more general?
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|
@ -1166,16 +1143,16 @@ END%%
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%%ANKI
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|
Basic
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|
Is $R = \{\langle a, b \rangle\}$ connected on set $\{a, b\}$?
|
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|
Back: Yes.
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|
*Why* isn't $R = \{\langle a, b \rangle\}$ connected on set $\{a, b\}$?
|
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|
Back: N/A. It is.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1722735199637-->
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END%%
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%%ANKI
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|
Basic
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|
|
|
|
Is $R = \{\langle a, a \rangle\}$ connected on set $\{a, b\}$?
|
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|
Back: No.
|
|
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|
|
*Why* isn't $R = \{\langle a, a \rangle\}$ connected on set $\{a, b\}$?
|
|
|
|
|
Back: Because $\langle a, b \rangle \not\in R$ and $\langle b, a \rangle \not\in R$.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1722735199645-->
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END%%
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|
@ -1212,14 +1189,6 @@ Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.
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<!--ID: 1722735199672-->
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END%%
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%%ANKI
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|
Basic
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|
|
Is $R = \{\langle a, b \rangle\}$ strongly connected on set $\{a, b\}$?
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|
|
Back: No.
|
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|
|
Reference: “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
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<!--ID: 1722735199678-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $R = \{\langle a, b \rangle\}$ strongly connected on set $\{a, b\}$?
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@ -1281,20 +1250,12 @@ END%%
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%%ANKI
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|
Basic
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|
|
Is $R = \{\langle 2, 3 \rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
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|
Back: Yes.
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|
*Why* isn't $R = \{\langle 2, 3 \rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
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Back: N/A. It is.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187602-->
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END%%
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%%ANKI
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Basic
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|
|
Is $R = \{\langle 2, 3 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
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|
Back: No.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1723245187605-->
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END%%
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%%ANKI
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Basic
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|
*Why* isn't $R = \{\langle 2, 3 \rangle, \langle 3, 5 \rangle\}$ trichotomous on $\{2, 3, 5\}$?
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|
@ -1303,14 +1264,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1723245187609-->
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END%%
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%%ANKI
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|
Basic
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|
|
Is $R = \{\langle 2, 2 \rangle\}$ trichotomous on $\{2\}$?
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|
|
Back: No.
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|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1723245187613-->
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END%%
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%%ANKI
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|
Basic
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|
*Why* isn't $R = \{\langle a, a \rangle\}$ trichotomous on $\{a\}$?
|
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|
|
@ -1398,9 +1351,180 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1723245187669-->
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END%%
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|
|
## Preorders
|
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|
|
$R$ is a **preorder on $A$** iff $R$ is a binary relation that is reflexive on set $A$ and transitive.
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%%ANKI
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Basic
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|
What is a preorder on $A$?
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Back: A binary relation reflexive on $A$ and transitive.
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|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723814834775-->
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END%%
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%%ANKI
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|
Basic
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|
|
Which of preorders or equivalence relations are the more general concept?
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|
Back: Preorders.
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|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723814834780-->
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END%%
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%%ANKI
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Basic
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|
*Why* are preorders named the way they are?
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|
|
Back: The name suggests its almost a partial order.
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|
|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723814834783-->
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END%%
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%%ANKI
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Basic
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|
*Why* isn't $R = \{\langle a, a \rangle\}$ a preorder?
|
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|
|
Back: N/A. The question must provide a reference set.
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|
|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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|
<!--ID: 1723814834790-->
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|
END%%
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%%ANKI
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|
Basic
|
|
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|
|
*Why* isn't $R = \{\langle a, a \rangle\}$ a preorder on $\{a\}$?
|
|
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|
|
Back: N/A. It is.
|
|
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|
|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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|
<!--ID: 1723814834793-->
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END%%
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%%ANKI
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|
Basic
|
|
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|
|
*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle, \langle a, c \rangle\}$ a preorder on $\{a, b, c\}$?
|
|
|
|
|
Back: Because $R$ isn't reflexive on $\{a, b, c\}$.
|
|
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|
|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723814834800-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle \}$ a preorder on $\{a, b\}$?
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Back: N/A. It is.
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Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723814834804-->
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END%%
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## Partial Orders
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$R$ is a **partial order on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, antisymmetric, and transitive.
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In other words, a partial order is an antisymmetric preorder.
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%%ANKI
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Basic
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What is a partial order on $A$?
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Back: A binary relation on $A$ that is reflexive on $A$, antisymmetric, and transitive.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108460-->
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END%%
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%%ANKI
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Basic
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Which of preorders and partial orders is the more general concept?
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Back: Preorders.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108468-->
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END%%
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%%ANKI
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Basic
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Which of partial orders and equivalence relations is the more general concept?
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Back: N/A.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108472-->
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END%%
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%%ANKI
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Cloze
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A preorder satisfying {antisymmetry} is a {partial order}.
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Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723816108477-->
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END%%
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%%ANKI
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Basic
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What two properties do partial orders and equivalence relations have in common?
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Back: Reflexivity and transitivity.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108482-->
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END%%
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%%ANKI
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Basic
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What property distinguishes partial orders from equivalence relations?
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Back: The former is antisymmetric whereas the latter is symmetric.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108487-->
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END%%
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%%ANKI
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Basic
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*Why* is a partial order named the way it is?
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Back: Not every pair of elements needs to be comparable.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108494-->
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END%%
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%%ANKI
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Basic
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Can a relation be both an equivalence relation and a partial order?
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Back: Yes.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108501-->
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END%%
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%%ANKI
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Basic
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Can a nonempty relation be both an equivalence relation and a partial order?
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Back: Yes.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108508-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle\}$ a partial order?
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Back: N/A. The question must provide a reference set.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108514-->
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END%%
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%%ANKI
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Basic
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|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, b \rangle\}$ a partial order on $\{a, b\}$?
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Back: N/A. It is.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108519-->
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END%%
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%%ANKI
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Basic
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|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a partial order on $\{a, b\}$?
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Back: N/A. It is.
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Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108524-->
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END%%
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%%ANKI
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|
Basic
|
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|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ a partial order on $\{a, b\}$?
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Back: It isn't antisymmetric.
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|
Reference: “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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<!--ID: 1723816108531-->
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END%%
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|
|
## Equivalence Relations
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|
|
Given relation $R$ and set $A$, $R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on $A$ that is reflexive on $A$, symmetric, and transitive.
|
|
|
|
|
$R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, symmetric, and transitive.
|
|
|
|
|
|
|
|
|
|
In other words, an equivalence relation is a symmetric preorder.
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|
%%ANKI
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|
Basic
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|
@ -1418,6 +1542,13 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1720967429853-->
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END%%
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%%ANKI
|
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|
|
|
Cloze
|
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|
|
A preorder satisfying {symmetry} is an {equivalence relation}.
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|
Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
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<!--ID: 1723814834787-->
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END%%
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%%ANKI
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|
Cloze
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|
An equivalence relation on $A$ is a {$2$}-ary relation on $A$.
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|
@ -1427,7 +1558,7 @@ END%%
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%%ANKI
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|
Basic
|
|
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|
|
Is $R = \{\langle a, a \rangle\}$ an equivalence relation?
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|
|
*Why* isn't $R = \{\langle a, a \rangle\}$ an equivalence relation?
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|
|
Back: N/A. The question must provide a reference set.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
<!--ID: 1720967429860-->
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|
@ -1435,20 +1566,12 @@ END%%
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%%ANKI
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|
Basic
|
|
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|
|
Is $R = \{\langle a, a \rangle\}$ an equivalence relation on $\{a\}$?
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|
Back: Yes.
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|
*Why* isn't $R = \{\langle a, a \rangle\}$ an equivalence relation on $\{a\}$?
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|
|
Back: N/A. It is.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
|
<!--ID: 1720967429864-->
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|
END%%
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%%ANKI
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|
Basic
|
|
|
|
|
Is $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ an equivalence relation on $\{a\}$?
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|
Back: No.
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|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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|
|
<!--ID: 1720967429868-->
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|
|
END%%
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|
|
%%ANKI
|
|
|
|
|
Basic
|
|
|
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ an equivalence relation on $\{a\}$?
|
|
|
|
|
@ -1473,6 +1596,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1720969371869-->
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|
END%%
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%%ANKI
|
|
|
|
|
Basic
|
|
|
|
|
*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ an equivalence relation on $\{a, b\}$?
|
|
|
|
|
Back: It isn't symmetric.
|
|
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|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
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|
|
<!--ID: 1723816108538-->
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|
|
END%%
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|
|
### Equivalence Classes
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|
|
The set $[x]_R$ is defined by $[x]_R = \{t \mid xRt\}$. If $R$ is an equivalence relation and $x \in \mathop{\text{fld}}R$, then $[x]_R$ is called the **equivalence class of $x$ (modulo $R$)**.
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|
@ -1874,4 +2005,6 @@ END%%
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|
|
* “Connected Relation,” in _Wikipedia_, July 14, 2024, [https://en.wikipedia.org/w/index.php?title=Connected_relation](https://en.wikipedia.org/w/index.php?title=Connected_relation&oldid=1234415201).
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|
|
* “Equivalence Relation,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Equivalence_relation](https://en.wikipedia.org/w/index.php?title=Equivalence_relation&oldid=1235801091).
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|
|
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
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|
|
* “Partially Ordered Set,” in _Wikipedia_, June 22, 2024, [https://en.wikipedia.org/w/index.php?title=Partially_ordered_set](https://en.wikipedia.org/w/index.php?title=Partially_ordered_set&oldid=1230452839).
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|
|
* “Partition of a Set,” in _Wikipedia_, June 18, 2024, [https://en.wikipedia.org/w/index.php?title=Partition_of_a_set](https://en.wikipedia.org/w/index.php?title=Partition_of_a_set&oldid=1229656401).
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|
* “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.org/w/index.php?title=Preorder](https://en.wikipedia.org/w/index.php?title=Preorder&oldid=1235839474).
|
