Set-builder notation notes.
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"_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f",
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"_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f",
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"_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b",
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"_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b",
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"set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
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"set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53",
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"set/index.md": "5302d73c3f54892d46413533b1eb04d0",
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"set/index.md": "9a270d08a96d0c8094ffb58671e50a12",
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"set/graphs.md": "55298be7241906cb6b61673cf0a2e709",
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"set/graphs.md": "55298be7241906cb6b61673cf0a2e709",
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"_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb",
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"_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb",
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"_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391",
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"_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391",
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@ -582,7 +582,7 @@
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"_journal/2024-07/2024-07-02.md": "489464ee47c3ba21307bfabae569ad29",
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"_journal/2024-07/2024-07-02.md": "489464ee47c3ba21307bfabae569ad29",
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"_journal/2024-07/2024-07-01.md": "7cffc27813fe7a7338e411d054ac3bd5",
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"_journal/2024-07/2024-07-01.md": "7cffc27813fe7a7338e411d054ac3bd5",
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"set/bags.md": "ba7990801734f411838d7b33e7ec0542",
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"set/bags.md": "ba7990801734f411838d7b33e7ec0542",
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"_journal/2024-07-07.md": "fbd2ad529330b632d65280c7fc221122",
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"_journal/2024-07-07.md": "9ee2d5007c34cc7ff681f3d9e998eca4",
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"_journal/2024-07/2024-07-06.md": "2b794e424985f0e7d4d899163ce5733c"
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"_journal/2024-07/2024-07-06.md": "2b794e424985f0e7d4d899163ce5733c"
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},
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},
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"fields_dict": {
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"fields_dict": {
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@ -8,4 +8,5 @@ title: "2024-07-07"
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- [ ] Sheet Music (10 min.)
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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- [ ] Korean (Read 1 Story)
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* Notes on [[bags]] and multigraphs.
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* Notes on [[bags]] and multigraphs.
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* Additional notes on set-builder notation and gotchas related to it.
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@ -96,6 +96,11 @@ Tags: adt::dynamic_set
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<!--ID: 1715432070083-->
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<!--ID: 1715432070083-->
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END%%
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END%%
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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 $10$:
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* Roster notation: $\{2, 3, 5, 7\}$
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* Set-builder notation: $\{x \mid x < 10 \land x \text{ is prime}\}$
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%%ANKI
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%%ANKI
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Basic
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Basic
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Define the set of prime numbers less than $10$ using abstraction.
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Define the set of prime numbers less than $10$ using abstraction.
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@ -206,6 +211,100 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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<!--ID: 1716494526284-->
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END%%
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END%%
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%%ANKI
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Basic
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How are members of the following set defined using extensionality and first-order logic? $$B = \{P(x) \mid \phi(x)\}$$
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Back: $\forall x, P(x) \in B \Leftrightarrow \phi(x)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624727-->
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END%%
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%%ANKI
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Basic
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How are members of the following set defined using extensionality and first-order logic? $$B = \{x \mid x < 5 \land x \text{ is prime}\}$$
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Back: $\forall x, x \in B \Leftrightarrow (x < 5 \land x \text{ is prime})$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624730-->
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END%%
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%%ANKI
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Cloze
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$P(x)$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624733-->
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END%%
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%%ANKI
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Cloze
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$\exists A \in B, uFx$ is equivalently written as $x \in$ {$\{v \mid \exists A \in B, uFv\}$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624735-->
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END%%
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%%ANKI
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Basic
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How is set $\{P(y) \mid y \in B\}$ interpreted?
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Back: As the set of $P(y)$ for all $y \in B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624736-->
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END%%
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%%ANKI
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Basic
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How many members are in set $\{P(y) \mid y \in B\}$?
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Back: As many as the number of unique $P(y)$ for each $y \in B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624737-->
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END%%
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%%ANKI
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Basic
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How is set $\{P(y) \mid \exists y \in B\}$ interpreted?
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Back: If $B$ is empty, the empty set. Otherwise as singleton $\{P(y)\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624738-->
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END%%
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%%ANKI
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Basic
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How many members are in set $\{P(y) \mid \exists y \in B\}$?
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Back: At most $1$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624739-->
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END%%
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%%ANKI
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Basic
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In set-builder notation, the left side of $\{\ldots \mid \ldots\}$ denotes what?
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Back: The members of the set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610010-->
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END%%
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%%ANKI
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Basic
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In set-builder notation, the right side of $\{\ldots \mid \ldots\}$ denotes what?
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Back: The entrance requirement.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610016-->
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END%%
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%%ANKI
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Basic
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How is set $\{v \mid \exists A \in B, v = A\}$ written more compactly?
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Back: $\{A \mid A \in B\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610022-->
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END%%
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%%ANKI
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Basic
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How is set $\{v \mid \exists A \in B, v \in A\}$ written more compactly?
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Back: N/A.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610028-->
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END%%
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## Extensionality
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## Extensionality
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If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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