Set-builder notation notes.

c-declarations
Joshua Potter 2024-07-07 13:45:15 -06:00
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- [ ] Korean (Read 1 Story) - [ ] Korean (Read 1 Story)
* Notes on [[bags]] and multigraphs. * Notes on [[bags]] and multigraphs.
* Additional notes on set-builder notation and gotchas related to it.

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<!--ID: 1715432070083--> <!--ID: 1715432070083-->
END%% END%%
Sets are often denoted using **roster notation** in which members are specified explicitly in a comma-delimited list surrounded by curly braces. Alternatively, **abstraction** (or **set-builder notation**) defines sets using an **entrance requirement**. Examples of the set of prime numbers less than $10$:
* Roster notation: $\{2, 3, 5, 7\}$
* Set-builder notation: $\{x \mid x < 10 \land x \text{ is prime}\}$
%%ANKI %%ANKI
Basic Basic
Define the set of prime numbers less than $10$ using abstraction. Define the set of prime numbers less than $10$ using abstraction.
@ -206,6 +211,100 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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END%% END%%
%%ANKI
Basic
How are members of the following set defined using extensionality and first-order logic? $$B = \{P(x) \mid \phi(x)\}$$
Back: $\forall x, P(x) \in B \Leftrightarrow \phi(x)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624727-->
END%%
%%ANKI
Basic
How are members of the following set defined using extensionality and first-order logic? $$B = \{x \mid x < 5 \land x \text{ is prime}\}$$
Back: $\forall x, x \in B \Leftrightarrow (x < 5 \land x \text{ is prime})$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624730-->
END%%
%%ANKI
Cloze
$P(x)$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624733-->
END%%
%%ANKI
Cloze
$\exists A \in B, uFx$ is equivalently written as $x \in$ {$\{v \mid \exists A \in B, uFv\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624735-->
END%%
%%ANKI
Basic
How is set $\{P(y) \mid y \in B\}$ interpreted?
Back: As the set of $P(y)$ for all $y \in B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624736-->
END%%
%%ANKI
Basic
How many members are in set $\{P(y) \mid y \in B\}$?
Back: As many as the number of unique $P(y)$ for each $y \in B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624737-->
END%%
%%ANKI
Basic
How is set $\{P(y) \mid \exists y \in B\}$ interpreted?
Back: If $B$ is empty, the empty set. Otherwise as singleton $\{P(y)\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624738-->
END%%
%%ANKI
Basic
How many members are in set $\{P(y) \mid \exists y \in B\}$?
Back: At most $1$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720369624739-->
END%%
%%ANKI
Basic
In set-builder notation, the left side of $\{\ldots \mid \ldots\}$ denotes what?
Back: The members of the set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720370610010-->
END%%
%%ANKI
Basic
In set-builder notation, the right side of $\{\ldots \mid \ldots\}$ denotes what?
Back: The entrance requirement.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720370610016-->
END%%
%%ANKI
Basic
How is set $\{v \mid \exists A \in B, v = A\}$ written more compactly?
Back: $\{A \mid A \in B\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720370610022-->
END%%
%%ANKI
Basic
How is set $\{v \mid \exists A \in B, v \in A\}$ written more compactly?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720370610028-->
END%%
## Extensionality ## Extensionality
If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$