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Notes on transitive sets.

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Joshua Potter 2024-09-19 20:04:19 -06:00
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* Notes on [[natural-numbers#Transitivity|transitive sets]].

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@ -538,6 +538,136 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1726364667688-->
END%%
## Transitivity
A set $A$ is said to be **transitive** iff every member of a member of $A$ is itself a member of $A$. We can equivalently express this using any of the following formulations:
* $x \in a \in A \Rightarrow x \in A$
* $\bigcup A \subseteq A$
* $a \in A \Rightarrow a \subseteq A$
* $A \subseteq \mathscr{P}A$
%%ANKI
Basic
What does it mean for $A$ to be a transitive set?
Back: Every member of a member of $A$ is itself a member of $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209150-->
END%%
%%ANKI
Basic
In what way is the term "transitive set" ambiguous?
Back: This term can also be used to describe a transitive relation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209152-->
END%%
%%ANKI
Cloze
A transitive {1:set} is to {2:membership} whereas a transitive {2:relation} is to {1:related}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209154-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$x \in a \in A$} $\Rightarrow$ {$x \in A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209155-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$\bigcup A$} $\subseteq$ {$A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209157-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$a \in A$} $\Rightarrow$ {$a \subseteq A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209158-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$A$} $\subseteq$ {$\mathscr{P} A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209159-->
END%%
%%ANKI
Basic
Is $\varnothing$ a transitive set?
Back: Yes.
<!--ID: 1726797209160-->
END%%
%%ANKI
Basic
*Why* isn't $\{0, 1\}$ a transitive set?
Back: N/A. It is.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209161-->
END%%
%%ANKI
Basic
*Why* isn't $\{1\}$ a transitive set?
Back: Because $0 \in 1$ but $0 \not\in \{1\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209163-->
END%%
%%ANKI
Basic
*Why* isn't $\{\varnothing\}$ a transitive set?
Back: N/A. It is.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209164-->
END%%
%%ANKI
Basic
*Why* isn't $\{\{\varnothing\}\}$ a transitive set?
Back: Because $\varnothing \in \{\varnothing\}$ but $\varnothing \not\in \{\{\varnothing\}\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209165-->
END%%
%%ANKI
Basic
Suppose $a$ is a transitive set. *Why* does $\bigcup a \cup a = a$?
Back: Because transitivity holds if and only if $\bigcup a \subseteq a$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209166-->
END%%
%%ANKI
Basic
Suppose $A \cup B = A$. What relation immediately follows?
Back: $B \subseteq A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209167-->
END%%
%%ANKI
Basic
Suppose $A \cap B = A$. What relation immediately follows?
Back: $B = A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797814900-->
END%%
%%ANKI
Cloze
$A$ is a transitive set iff {$\bigcup$}$A^+ =$ {$A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1726797209168-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

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<!--ID: 1720969371859-->
END%%
%%ANKI
Basic
The term "transitive" is used to describe what kind of mathematical object?
Back: Relations.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1721694448736-->
END%%
## Connected
A binary relation $R$ on set $A$ is said to be **connected** if for any *distinct* $x, y \in A$, either $xRy$ or $yRx$. The relation is **strongly connected** if for *all* $x, y \in A$, either $xRy$ or $yRx$.