11 KiB
title | TARGET DECK | FILE TAGS | tags | ||
---|---|---|---|---|---|
Natural Numbers | Obsidian::STEM | set::nat |
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Overview
The standard way of representing the natural numbers is as follows:
0 = \varnothing
1 = \{0\} = \{\varnothing\}
2 = \{0, 1\} = \{\varnothing, \{\varnothing\}\}
\ldots
That is, each natural number corresponds to the set of natural numbers smaller than it.
%%ANKI
Basic
How is the number 0
represented as a set?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is the number 1
represented as a set?
Back: \{0\} = \{\varnothing\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is the number 2
represented as a set?
Back: \{0, 1\} = \{\varnothing, \{\varnothing\}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Who came up with the standard set representation of natural numbers? Back: John von Neumann. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider the set representation of n \in \mathbb{N}
. How many members does n
have?
Back: n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider the set representation of n \in \mathbb{N}
. What are the members of n
?
Back: 0
, 1
, \ldots
, n - 1
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \mathbb{N}
. Why is n \in n + 1
?
Back: n + 1
is a set containing all preceding natural numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \mathbb{N}
. Why is n \subseteq n + 1
?
Back: n
and n + 1
are sets containing all their preceding natural numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Inductive Sets
For any set a
, its successor a^+
is defined as $a^+ = a \cup \{a\}
$
%%ANKI
Basic
How is the successor of a set a
denoted?
Back: a^+
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is the successor of a set a
defined?
Back: As a^+ = a \cup \{a\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Set \{a, b\}^+
equals what other set?
Back: \{a, b, \{a, b\}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Set \{a\}^+
equals what other set?
Back: \{a, \{a\}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Set \{a, \{a, b\}, \{a, b, c\}\}
can be written as the successor of what set?
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Set \{a, b, \{a, b\}\}
can be written as the successor of what set?
Back: \{a, b\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Set \{a, \{a, b\}\}
can be written as the successor of what set?
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Set \{a, \{a, b\}, \{a, \{a, b\}\}\}
can be written as the successor of what set?
Back: \{a, \{a, b\}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
If n \in \mathbb{N}
then n \in n + 1
. What analagous statement holds for arbitrary set a
?
Back: a \in a^+
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
If n \in \mathbb{N}
then n \subseteq n + 1
. What analagous statement holds for arbitrary set a
?
Back: a \subseteq a^+
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
A set A
is inductive if and only if \varnothing \in A
and \forall a \in A, a^+ \in A
.
%%ANKI
Basic
What does it mean for a set A
to be closed under successor?
Back: If a \in A
, then a^+ \in A
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Write "set B
is closed under successor" in FOL.
Back: \forall b \in B, b^+ \in B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does it mean for a set A
to be inductive?
Back: \varnothing \in A
and A
is closed under successor.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
A set A
is inductive iff {\varnothing \in A
} and {A
is closed under successor}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic An inductive set is closed under what operation? Back: Successor. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What set is the "seed" of an inductive set?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let a \in A
where A
is an inductive set. What other members must belong to A
?
Back: a^+
, a^{++}
, \ldots
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What natural number corresponds to \varnothing^{+++}
?
Back: 3
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What natural number corresponds to \varnothing
?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
A natural number is a set that belongs to every inductive set.
%%ANKI Basic How is a natural number defined in set theory? Back: As a set belonging to every inductive set. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What greek letter is used to denote the set of natural numbers?
Back: \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
In set theory, \omega
denotes what set?
Back: The natural numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the smallest inductive set?
Back: \omega
, i.e. the set of natural numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How might \omega
be defined as an intersection of classes?
Back: \omega = \bigcap\,\{A \mid A \text{ is inductive}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose n \in \omega
. What other sets must n
be a member of?
Back: Every other inductive set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What can be said about a subset of \omega
?
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%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).
END%%
%%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).
END%%
%%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).
END%%
%%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).
END%%
%%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).
END%%
%%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).
END%%
%%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).
END%%
%%ANKI
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).
END%%
%%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).
END%%
Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).