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| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Axioms | Obsidian::STEM | set |
|
Overview
Enderton describes ten different axioms in total which serve as the foundation of our set theory.
Extensionality
If two sets have exactly the same members, then they are equal: \forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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Basic
What does the extensionality axiom state?
Back: If two sets have exactly the same members, then they are equal.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the extensionality axiom expressed using first-order logic?
Back: \forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
The following encodes which set theory axiom? \forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
Back: The extensionality axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic How many sets exist with no members? Back: Exactly one. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Which set theory axiom proves uniqueness of \varnothing?
Back: The extensionality axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Empty Set Axiom
There exists a set having no members: \exists B, \forall x, x \not\in B$$
%%ANKI Basic What does the empty set axiom state? Back: There exists a set having no members. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the empty set axiom expressed using first-order logic?
Back: \exists B, \forall x, x \not\in B$$
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
The following encodes which set theory axiom? \exists B, \forall x, x \not\in B$$
Back: The empty set axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Which set theory axiom proves existence of \varnothing?
Back: The empty set axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What two properties ensures definition \varnothing is well-defined?
Back: The empty set exists and is unique.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the empty set defined using set-builder notation?
Back: \{x \mid x \neq x\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Pairing Axiom
For any sets u and v, there exists a set having as members just u and v: \forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)
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Basic
What does the pairing axiom state?
Back: For any sets u and v, there exists a set having as members just u and v.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the pairing axiom expressed using first-order logic?
Back: \forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
The following encodes which set theory axiom? \forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)
Back: The pairing axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Which set theory axiom proves existence of set \{x, y\} where x \neq y?
Back: The pairing axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Which set theory axiom proves existence of set \{x\}?
Back: The pairing axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
For sets u and v, what name is given to set \{u, v\}?
Back: The pair set of u and v.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic In set theory, what does a singleton refer to? Back: A set with exactly one member. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic What set theory axiom is used to prove existence of singletons? Back: The pairing axiom. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the pair set \{u, v\} defined using set-builder notation?
Back: \{x \mid x = u \lor x = v\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Union Axiom
Preliminary Form
For any sets a and b, there exists a set whose members are those sets belonging either to a or to b (or both): \forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)
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Basic
What does the union axiom (preliminary form) state?
Back: For any sets a and b, there exists a set whose members are all in either a or b.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the union axiom (preliminary form) expressed using first-order logic?
Back: \forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
The following encodes which set theory axiom? \forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)
Back: The union axiom (preliminary form).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the union of sets a and b denoted?
Back: a \cup b
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What two set theory axioms prove existence of e.g. \{x_1, x_2, x_3\}?
Back: The pairing axiom and union axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the union of set a and b defined using set-builder notation?
Back: \{x \mid x \in a \lor x \in b\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Power Set Axiom
For any set a, there is a set whose members are exactly the subsets of a: \forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)
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Basic
What does the power set axiom state?
Back: For any set a, there exists a set whose members are exactly the subsets of a.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the power set axiom expressed using first-order logic?
Back: \forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
The following encodes which set theory axiom? \forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)
Back: The power set axiom.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is x \subseteq a rewritten using first-order logic and \in?
Back: \forall t, t \in x \Rightarrow t \in a
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the power set of set a denoted?
Back: \mathscr{P}{a}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the power set of set a defined using set-builder notation?
Back: \{x \mid x \subseteq a\}
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).