For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
%%ANKI
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).
<!--ID: 1715649734337-->
END%%
%%ANKI
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).
<!--ID: 1715649734341-->
END%%
%%ANKI
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).
<!--ID: 1715649734346-->
END%%
%%ANKI
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).
<!--ID: 1715649734351-->
END%%
%%ANKI
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).
<!--ID: 1715649734357-->
END%%
%%ANKI
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).
<!--ID: 1715688034322-->
END%%
%%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).
<!--ID: 1715688034325-->
END%%
%%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).
For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both): $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
%%ANKI
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).
<!--ID: 1715688034333-->
END%%
%%ANKI
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).
<!--ID: 1715688034337-->
END%%
%%ANKI
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).
<!--ID: 1715688034341-->
END%%
%%ANKI
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).
For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
%%ANKI
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).
<!--ID: 1715688034356-->
END%%
%%ANKI
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).
<!--ID: 1715688034361-->
END%%
%%ANKI
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).
<!--ID: 1715688034368-->
END%%
%%ANKI
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).
<!--ID: 1715688034375-->
END%%
%%ANKI
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).