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