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Additional notes on equivalence relations.

c-declarations
Joshua Potter 2024-07-14 09:03:36 -06:00
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* Notes on infinite Cartesian products and their relation to the axiom of choice.
* Notes on [[set#Cartesian Product|infinite Cartesian products]] and their relation to the [[set/index#Infinite Cartesian Product Form|axiom of choice]].
* Initial notes on [[relations#Equivalence Relations|equivalence relations]].

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<!--ID: 1718329620208-->
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 **reflexive on $A$** if $xRx$ for all $x \in A$.
* $R$ is **symmetric** if whenever $xRy$, then $yRx$.
* $R$ is **transitive** if whenever $xRy$ and $yRz$, then $xRz$.
%%ANKI
Cloze
Binary relation $R$ is {reflexive on $A$} iff {$xRx$ for all $x \in A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429790-->
END%%
%%ANKI
Basic
Why is it incorrect to ask if $R$ is reflexive?
Back: We have to ask if $R$ is reflexive on some reference set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429800-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive?
Back: N/A. The question must provide a reference set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429804-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$?
Back: N/A. We must ask if $R$ is reflexive on a set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429808-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $\{a\}$?
Back: Yes.
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.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429817-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, *why* isn't $R$ 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: 1720967429820-->
END%%
%%ANKI
Cloze
If $xRx$ for all $x \in A$, $R$ is said to be reflexive {on} $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429824-->
END%%
%%ANKI
Cloze
Binary relation $R$ is {symmetric} iff {$xRy \Rightarrow yRx$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429828-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ symmetric?
Back: No.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429832-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, what additional member(s) must be added to make $R$ symmetric?
Back: $\langle b, a \rangle$ and $\langle c, b \rangle$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429835-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, b \rangle\}$, which of reflexivity (on $\{a, b\}$), symmetry, and transitivity does $R$ exhibit?
Back: Reflexivity on $\{a, b\}$ and symmetry.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429839-->
END%%
%%ANKI
Cloze
Binary relation $R$ is {transitive} iff {$xRy \land yRz \Rightarrow xRz$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429843-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, b \rangle, \langle b, c \rangle\}$, is $R$ transitive?
Back: No.
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
What is an equivalence relation on $A$?
Back: A binary relation on $A$ that is reflexive on $A$, symmetric, and transitive.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429853-->
END%%
%%ANKI
Cloze
An equivalence relation on $A$ is a {$2$}-ary relation on $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429857-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle\}$, is $R$ 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-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle\}$, is $R$ an equivalence relation on $\{a\}$?
Back: Yes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429864-->
END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ 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 $\{a\}$?
Back: $R$ is neither symmetric nor transitive.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720967429873-->
END%%
%%ANKI
Basic
Which of symmetric relations and transitive relations is more general?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720969371859-->
END%%
%%ANKI
Basic
Which of equivalence relations on $A$ and symmetric relations is more general?
Back: Symmetric relations.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720969371866-->
END%%
%%ANKI
Basic
Which of binary relations on $A$ and equivalence relations on $A$ is more general?
Back: Binary relations on $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720969371869-->
END%%
## Bibliography
* “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).