Strict preorders/partial orders.
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},
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"fields_dict": {
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"Basic": [
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---
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title: "2024-08-17"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Notes on strict preorders/partial orders.
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@ -703,7 +703,7 @@ END%%
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%%ANKI
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Basic
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What is it that universal hashing makes impossible?
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Back: The ability of an adversary forcing the worst-case running time of hash table operations.
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Back: Forcing the worst-case running time of hash table operations.
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Tags: hashing::random hashing::universal
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<!--ID: 1722080163399-->
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END%%
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@ -1358,7 +1358,7 @@ $R$ is a **preorder on $A$** iff $R$ is a binary relation that is reflexive on s
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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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Back: A binary relation on $A$ that is 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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@ -1411,11 +1411,82 @@ Reference: “Preorder,” in _Wikipedia_, July 21, 2024, [https://en.wikipedia.
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<!--ID: 1723814834804-->
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END%%
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A **strict preorder** replaces reflexivity with irreflexivity. That is, $R$ is a strict preorder on $A$ iff $R$ is a binary relation on set $A$ that is irreflexive on $A$ and transitive.
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%%ANKI
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Basic
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What distinguishes a preorder from a strict preorder?
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Back: Strict preorders are irreflexive.
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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: 1723902729046-->
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END%%
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%%ANKI
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Basic
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What is a strict preorder on $A$?
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Back: A binary relation on $A$ that is irreflexive 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: 1723902729097-->
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END%%
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%%ANKI
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Basic
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What makes a strict preorder more strict than a non-strict preorder?
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Back: Strict preorders do not allow relating members to themselves.
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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: 1723902729104-->
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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 strict preorder on $\{a\}$?
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Back: $R$ isn't irreflexive.
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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: 1723902729111-->
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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 strict preorder on $\{a, b, c\}$?
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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: 1723902729117-->
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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 strict preorder on $\{a, b\}$?
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Back: $R$ isn't irreflexive.
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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: 1723902729122-->
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END%%
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%%ANKI
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Cloze
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A {1:strict} preorder is equivalent to a {1:strict} 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: 1723902729128-->
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END%%
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%%ANKI
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Basic
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*Why* is a strict preorder also a strict partial order?
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Back: Irreflexivity and transitivity imply asymmetry (and antisymmetry).
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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: 1723902729134-->
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END%%
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%%ANKI
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Basic
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What equivalence in order theory serves as a mnemonic for "irreflexivity and transitivity imply asymmetry"?
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Back: A strict preorder is equivalent to a strict 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: 1723902729140-->
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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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$R$ 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.
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%%ANKI
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Basic
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@ -1506,7 +1577,7 @@ 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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*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle\}$ a partial order on $\{a, b, c\}$?
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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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@ -1514,12 +1585,84 @@ 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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*Why* isn't $R = \{\langle a, a \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ a partial order on $\{a, b, c\}$?
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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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A **strict partial order** replaces reflexivity with irreflexivity. That is, $R$ is a strict partial order on $A$ iff $R$ is a binary relation on set $A$ that is irreflexive on $A$, antisymmetric, and transitive.
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%%ANKI
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Basic
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What distinguishes a partial order from a strict partial order?
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Back: Strict partial orders are irreflexive.
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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: 1723902024372-->
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END%%
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%%ANKI
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Basic
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What is a strict partial order on $A$?
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Back: A binary relation on $A$ that is irreflexive 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: 1723902024375-->
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END%%
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%%ANKI
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Basic
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What makes a strict partial order more strict than a non-strict partial order?
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Back: Strict partial orders do not allow relating members to themselves.
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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: 1723902729147-->
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END%%
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%%ANKI
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Cloze
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Operator {$<$} typically denote a {strict} partial order.
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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: 1723902024378-->
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END%%
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%%ANKI
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Cloze
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Operator {$\leq$} typically denote a {non-strict} partial order.
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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: 1723902024382-->
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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 strict 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: 1723902024385-->
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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 strict partial order on $\{a, b, c\}$?
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Back: Because it isn't irreflexive.
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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: 1723902024388-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $R = \{\langle a, c \rangle, \langle b, c \rangle\}$ a strict partial order on $\{a, b, c\}$?
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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: 1723902024391-->
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END%%
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%%ANKI
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Basic
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*Why* isn't $R = \{\langle a, b \rangle, \langle b, c \rangle, \langle c, b \rangle\}$ a strict partial order on $\{a, b\}$?
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Back: It is neither antisymmetric nor 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: 1723902024394-->
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END%%
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## Equivalence Relations
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$R$ is an **equivalence relation on $A$** iff $R$ is a binary relation on set $A$ that is reflexive on $A$, symmetric, and transitive.
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