7.8 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Natural Numbers | Obsidian::STEM | set::nat |
|
Overview
The standard way of representing the natural numbers is as follows:
0 = \varnothing1 = \{0\} = \{\varnothing\}2 = \{0, 1\} = \{\varnothing, \{\varnothing\}\}\ldots
That is, each natural number corresponds to the set of natural numbers smaller than it.
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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).
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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).
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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).
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%%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).
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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).
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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).
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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).
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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).
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Inductive Sets
For any set a, its successor a^+ is defined as $a^+ = a \cup \{a\}$
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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).
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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).
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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).
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Basic
Set \{a\}^+ equals what other set?
Back: \{a, \{a\}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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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).
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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%%
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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%%
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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%%
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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).
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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).
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A set A is inductive if and only if \varnothing \in A and \forall a \in A, a^+ \in A.
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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).
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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).
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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).
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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).
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%%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).
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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).
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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).
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Basic
What natural number corresponds to \varnothing^{+++}?
Back: 3
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What natural number corresponds to \varnothing?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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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).
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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).
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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).
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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).
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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).
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Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).