Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What can be said about an inductive subset of $\omega$?
Back: It must coincide with $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Why must every inductive subset of $\omega$ coincide with $\omega$?
Back: Because $\omega$ is the smallest inductive set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What does the induction principle for $\omega$ state?
Back: Every inductive subset of $\omega$ coincides with $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What name is given to the principle, "every inductive subset of $\omega$ coincides with $\omega$?"
Back: The induction principle for $\omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Inductive sets correspond to what kind of proof method?
Back: Proof by induction.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
Prove $P(n)$ is true for all $n \in \mathbb{N}$ using induction. What set do we prove is inductive?
Back: $\{n \in \mathbb{N} \mid P(n)\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
*How* are inductive sets and proof by induction related?
Back: An induction proof corresponds to proving a related set is inductive.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
What inductive set do we construct to prove the following by induction? $$\text{Every natural number is nonnegative}$$
Back: $\{n \in \omega \mid 0 \leq n\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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%%ANKI
Basic
What inductive set do we construct to prove the following by induction? $$\text{Every nonzero natural number is the successor of another natural number}$$
Back: $\{n \in \omega \mid n = 0 \lor (\exists m \in \omega, n = m^+)\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
A **Peano system** is a triple $\langle N, S, e \rangle$ consisting of a set $N$, a function $S \colon N \rightarrow N$, and a member $e \in N$ such that the following three conditions are met:
* $e \not\in \mathop{\text{ran}}{S}$;
* $S$ is one-to-one;
* Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself.
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).
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%%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).
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%%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).
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%%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).
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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).
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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).
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%%ANKI
Basic
Is $\varnothing$ a transitive set?
Back: Yes.
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%%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).
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%%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).
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%%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).
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%%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).
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%%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).
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%%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).
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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).
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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).
