40 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Algebra of Sets | Obsidian::STEM | algebra::set set |
|
Overview
The study of the operations of union (\cup), intersection (\cap), and set difference (-), together with the inclusion relation (\subseteq), goes by the algebra of sets.
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Basic
What three operators make up the algebra of sets?
Back: \cup, \cap, and -.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What relation is relevant in the algebra of sets?
Back: \subseteq
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Symmetric Difference
Define the symmetric difference of sets A and B as $A \mathop{\triangle} B = (A - B) \cup (B - A)$
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Basic
What two operators are used in the definition of the symmetric difference?
Back: \cup and -.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How is the symmetric difference of sets A and B denoted?
Back: A \mathop{\triangle} B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
How is A \mathop{\triangle} B defined?
Back: As (A - B) \cup (B - A).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cartesian Product
Given two sets A and B, the Cartesian product A \times B is defined as: $A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$
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Basic
How is the Cartesian product of A and B denoted?
Back: A \times B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Using ordered pairs, how is A \times B defined?
Back: \{\langle x, y \rangle \mid x \in A \land y \in B\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic Who is attributed the representation of points in a plane? Back: René Descartes. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic Why is the Cartesian product named the way it is? Back: It is named after René Descartes. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
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Basic
Suppose x, y \in A. What set, derived from A, is \langle x, y \rangle a member of?
Back: \mathscr{P}\mathscr{P}A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
{x \in A} iff {\{x\} \subseteq A} iff {\{x\} \in \mathscr{P}A}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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We can also form (something like) the Cartesian product of infinitely many sets, provided that the sets are suitably indexed. Let I be an index set and H a function whose domain includes I. Define \bigtimes_{i \in I} H(i) = {f \mid f \text{ is a function with domain } I \text{ and } \forall i \in I, f(i) \in H(i)}
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Basic
What kind of mathematical object is I in \bigtimes_{i \in I} H(i)?
Back: A set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What kind of mathematical object is H in \bigtimes_{i \in I} H(i)?
Back: A function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the domain of H in \bigtimes_{i \in I} H(i)?
Back: Some superset of I.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What is the range of H in \bigtimes_{i \in I} H(i)?
Back: Some superset of \{H(i) \mid i \in I\}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Let I be an index set and H a function such that I \subseteq \mathop{\text{dom}}H. How is \bigtimes_{i \in I} H(i) defined?
Back: \bigtimes_{i \in I} H(i) = \{ f \mid f \text{ is a function with domain } I \text { and } \forall i \in I, f(i) \in H(i) \}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What kind of mathematical object is h \in \bigtimes_{i \in I} H(i)?
Back: A function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Let f \in \bigtimes_{i \in I} H(i). What is the domain of f?
Back: I
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Let f \in \bigtimes_{i \in I} H(i). What is the codomain of f?
Back: \bigcup_{i \in I} H(i)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Given arbitrary sets A and B, what index set I and function H satisfies A \times B = \bigtimes_{i \in I} H(i)?
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Why can't A \times B be rewritten with \bigtimes_{i \in I} H(i) assuming suitable I and H?
Back: The former is a set of ordered pairs. The latter is a set of functions.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Assume AoC and H(j) = \varnothing for some j \in I. What does \bigtimes_{i \in I} H(i) evaluate to?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
When does \bigtimes_{i \in I} H(i) = \varnothing?
Back: When there exists some i \in I such that H(i) = \varnothing.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Assume AoC and H(j) \neq \varnothing for all j \in I. What does \bigtimes_{i \in I} H(i) evaluate to?
Back: A non-empty set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
The following is likely a diagram of what?
!
Back: A member of \bigtimes_{i \in \omega} H(i).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Suppose H(i) \neq \varnothing for all i \in I. When is \bigtimes_{i \in I} H(i) \neq \varnothing?
Back: When AoC is included in our formal system.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Laws
Commutative Laws
For any sets A and B, \begin{align*} A \cup B & = B \cup A \ A \cap B & = B \cap A \end{align*}
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Basic
The commutative laws of the algebra of sets apply to what operators?
Back: \cup and \cap
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What does the union commutative law state?
Back: For any sets A and B, A \cup B = B \cup A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What does the intersection commutative law state?
Back: For any sets A and B, A \cap B = B \cap A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Is the Cartesian product commutative? Back: No. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
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%%ANKI Basic Why isn't the Cartesian product commutative? Back: Because the Cartesian product comprises of ordered pairs. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
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Basic
Suppose A \neq \varnothing and B \neq \varnothing. When does A \times B = B \times A?
Back: When A = B.
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Basic
Suppose A \neq \varnothing and A \neq B. When does A \times B = B \times A?
Back: When B = \varnothing.
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Basic
Under what two conditions is A \times B = B \times A?
Back: A = B or either set is the empty set.
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
Associative Laws
For any sets A and B, \begin{align*} A \cup (B \cup C) & = (A \cup B) \cup C \ A \cap (B \cap C) & = (A \cap B) \cap C \end{align*}
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Basic
The associative laws of the algebra of sets apply to what operators?
Back: \cup and \cap
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What does the union associative law state?
Back: For any sets A, B, and C, A \cup (B \cup C) = (A \cup B) \cup C.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What does the intersection associative law state?
Back: For any sets A, B, and C, A \cap (B \cap C) = (A \cap B) \cap C.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Is the Cartesian product associative? Back: No. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
%%ANKI Basic Why isn't the Cartesian product associative? Back: The association of parentheses defines the nesting of the ordered pairs. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
Distributive Laws
For any sets A, B, and C, \begin{align*} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{align*}
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Basic
The distributive laws of the algebra of sets apply to what operators?
Back: \cup and \cap
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
The distributive law states {A \cap (B \cup C)} = {(A \cap B) \cup (A \cap C)}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
The distributive law states {A \cup (B \cap C)} = {(A \cup B) \cap (A \cup C)}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What concept in set theory relates the algebra of sets to boolean algebra? Back: Membership. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What two equalities relates A \cup B with a \lor b?
Back: a = (x \in A) and b = (x \in B).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What two equalities relates A \cap B with a \land b?
Back: a = (x \in A) and b = (x \in B).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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More generally, for any sets A and \mathscr{B}, \begin{align*} A \cup \bigcap \mathscr{B} & = \bigcap, {A \cup X \mid X \in \mathscr{B}}, \text{ for } \mathscr{B} \neq \varnothing \ A \cap \bigcup \mathscr{B} & = \bigcup, {A \cap X \mid X \in \mathscr{B}} \end{align*}
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Basic
What is the generalization of identity A \cap (B \cup C) = (A \cap B) \cup (A \cap C)?
Back: A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the generalization of identity A \cup (B \cap C) = (A \cup B) \cap (A \cup C)?
Back: A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\} for \mathscr{B} \neq \varnothing.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
Assuming \mathscr{B} \neq \varnothing, the distributive law states {A \cup \bigcap \mathscr{B}} = {\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cloze
The distributive law states {A \cap \bigcup \mathscr{B}} = {\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
How is set \{A \cup X \mid X \in \mathscr{B}\} pronounced?
Back: The set of all A \cup X such that X \in \mathscr{B}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the specialization of identity A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}?
Back: A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the specialization of identity A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}?
Back: A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Does \bigcup\, \{A \cap X \mid X \in \mathscr{B}\} get smaller or larger as \mathscr{B} gets larger?
Back: Larger.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Does \bigcup\, \{A \cap X \mid X \in \mathscr{B}\} get smaller or larger as \mathscr{B} gets smaller?
Back: Smaller.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Does \bigcap\, \{A \cup X \mid X \in \mathscr{B}\} get smaller or larger as \mathscr{B} \neq \varnothing gets larger?
Back: Smaller.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Does \bigcap\, \{A \cup X \mid X \in \mathscr{B}\} get smaller or larger as \mathscr{B} \neq \varnothing gets smaller?
Back: Larger.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
For any sets A, B, and C, \begin{align*} A \times (B \cap C) & = (A \times B) \cap (A \times C) \ A \times (B \cup C) & = (A \times B) \cup (A \times C) \ A \times (B - C) & = (A \times B) - (A \times C) \end{align*}
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Basic
Which of the algebra of sets operators does the Cartesian product distributive over?
Back: \cap, \cup, and -
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Basic
What distributivity rule is satisfied by \cap and \times?
Back: A \times (B \cap C) = (A \times B) \cap (A \times C)
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Cloze
The Cartesian product satisfies distributivity: {A \times (B \cap C)} = {(A \times B) \cap (A \times C)}.
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Basic
What distributivity rule is satisfied by \cup and \times?
Back: A \times (B \cup C) = (A \times B) \cup (A \times C)
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Cloze
The Cartesian product satisfies distributivity: {A \times (B \cup C)} = {(A \times B) \cup (A \times C)}.
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Basic
What distributivity rule is satisfied by - and \times?
Back: A \times (B - C) = (A \times B) - (A \times C)
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
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Cloze
The Cartesian product satisfies distributivity: {A \times (B - C)} = {(A \times B) - (A \times C)}.
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
END%%
In addition, \begin{align*} A \times \bigcup \mathscr{B} & = \bigcup, {A \times X \mid X \in \mathscr{B}} \ A \times \bigcap \mathscr{B} & = \bigcap, {A \times X \mid X \in \mathscr{B}} \end{align*}
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Basic
What is the generalization of identity A \times (B \cup C) = (A \times B) \cup (A \times C)?
Back: A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the specialization of identity A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}?
Back: A \times (B \cap C) = (A \times B) \cap (A \times C)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the generalization of identity A \times (B \cap C) = (A \times B) \cap (A \times C)?
Back: A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the specialization of identity A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}?
Back: A \times (B \cup C) = (A \times B) \cup (A \times C)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
De Morgan's Laws
For any sets A, B, and C, \begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}
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Basic
The De Morgan's laws of the algebra of sets apply to what operators?
Back: \cup, \cap, and -
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
De Morgan's law states that {C - (A \cup B)} = {(C - A) \cap (C - B)}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
De Morgan's law states that {C - (A \cap B)} = {(C - A) \cup (C - B)}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
For their respective De Morgan's laws, {-} is to the algebra of sets whereas {\neg} is to boolean algebra.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
For their respective De Morgan's laws, {\cup} is to the algebra of sets whereas {\lor} is to boolean algebra.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
For their respective De Morgan's laws, {\cap} is to the algebra of sets whereas {\land} is to boolean algebra.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
More generally, for any sets C and \mathscr{A} \neq \varnothing, \begin{align*} C - \bigcup \mathscr{A} & = \bigcap, {C - X \mid X \in \mathscr{A}} \ C - \bigcap \mathscr{A} & = \bigcup, {C - X \mid X \in \mathscr{A}} \end{align*}
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Basic
What is the generalization of identity C - (A \cup B) = (C - A) \cap (C - B)?
Back: C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\} for \mathscr{A} \neq \varnothing.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the generalization of identity C - (A \cap B) = (C - A) \cup (C - B)?
Back: C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\} for \mathscr{A} \neq \varnothing.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Cloze
For \mathscr{A} \neq \varnothing, De Morgan's law states that {C - \bigcap \mathscr{A}} = {\bigcup\, \{C - X \mid X \in \mathscr{A}\}}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the specialization of identity C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}?
Back: C - (A \cup B) = (C - A) \cap (C - B)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the specialization of identity C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}?
Back: C - (A \cap B) = (C - A) \cup (C - B)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Which law of the algebra of sets is represented by e.g. C - (A \cup B) = (C - A) \cap (C - B)?
Back: De Morgan's Law.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
For \mathscr{A} \neq \varnothing, De Morgan's law states that {C - \bigcup \mathscr{A}} = {\bigcap\, \{C - X \mid X \in \mathscr{A}\}}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Why does identity C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\} fail when \mathscr{A} = \varnothing?
Back: The RHS evaluates to class \bigcap \varnothing.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Why does identity C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\} fail when \mathscr{A} = \varnothing?
Back: \bigcap \mathscr{A} is undefined.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Does \bigcap\, \{C - X \mid X \in \mathscr{A}\} get smaller or larger as \mathscr{A} \neq \varnothing gets larger?
Back: Smaller.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Does \bigcap\, \{C - X \mid X \in \mathscr{A}\} get smaller or larger as \mathscr{A} \neq \varnothing gets smaller?
Back: Larger.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Does \bigcup\, \{C - X \mid X \in \mathscr{A}\} get smaller or larger as \mathscr{A} \neq \varnothing gets larger?
Back: Larger.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Does \bigcup\, \{C - X \mid X \in \mathscr{A}\} get smaller or larger as \mathscr{A} \neq \varnothing gets smaller?
Back: Smaller.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Monotonicity
Let A, B, and C be arbitrary sets. Then
A \subseteq B \Rightarrow A \cup C \subseteq B \cup C,A \subseteq B \Rightarrow A \cap C \subseteq B \cap C,A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B
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Basic
The monotonicity properties of \subseteq are what kind of propositional logical statement?
Back: Implications.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
What is the shared antecedent of the monotonicity properties of \subseteq?
Back: A \subseteq B for some sets A and B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Given sets A, B, and C, state the monotonicity property of \subseteq related to the \cup operator.
Back: A \subseteq B \Rightarrow A \cup C \subseteq B \cup C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given sets A, B, and C, state the monotonicity property of \subseteq related to the \cap operator.
Back: A \subseteq B \Rightarrow A \cap C \subseteq B \cap C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given sets A and B, state the monotonicity property of \subseteq related to the \bigcup operator.
Back: A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why are the monotonicity properties of \subseteq named the way they are?
Back: The ordering of operands in the antecedent are preserved in the consequent.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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In addition,
A \subseteq B \Rightarrow A \times C \subseteq B \times C
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Basic
What monotonicity property does the Cartesian product satisfy?
Back: A \subseteq B \Rightarrow A \times C \subseteq B \times C
Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305.
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Antimonotonicity
Let A, B, and C be arbitrary sets. Then
A \subseteq B \Rightarrow C - B \subseteq C - A,\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A
%%ANKI
Basic
What kind of propositional logical statements are the antimonotonicity properties of \subseteq?
Back: Implications.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What is the shared antecedent of the antimonotonicity properties of \subseteq?
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cloze
{1:Monotonicity} of \subseteq is to {2:\bigcup} whereas {2:antimonotonicity} of \subseteq is to {1:\bigcap}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Why are the antimonotonicity properties of \subseteq named the way they are?
Back: The ordering of operands in the antecedent are reversed in the consequent.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given sets A and B, state the antimonotonicity property of \subseteq related to the \bigcap operator.
Back: \varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Given sets A, B, and C, state the antimonotonicity property of \subseteq related to the - operator.
Back: A \subseteq B \Rightarrow C - B \subseteq C - A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why do we need the empty set check in \varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A?
Back: \bigcap A is not a set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cancellation Laws
Let A, B, and C be sets. If A \neq \varnothing,
(A \times B = A \times C) \Rightarrow B = C(B \times A = C \times A) \Rightarrow B = C
%%ANKI
Basic
What is the left cancellation law of the Cartesian product?
Back: If A \neq \varnothing then (A \times B = A \times C) \Rightarrow B = C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
(A \times B = A \times C) \Rightarrow B = C is always true if what condition is satisfied?
Back: A \neq \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the right cancellation law of the Cartesian product?
Back: If A \neq \varnothing then (B \times A = C \times A) \Rightarrow B = C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
(B \times A = C \times A) \Rightarrow B = C is always true if what condition is satisfied?
Back: A \neq \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Index Sets
Let I be a set, called the index set. Let F be a functions whose domain includes I. Then we define \bigcup_{i \in I} F(i) = \bigcup,{F(i) \mid i \in I}
and, if I \neq \varnothing, \bigcap_{i \in I} F(i) = \bigcap, {F(i) \mid i \in I}
%%ANKI
Basic
What name does I go by in expression \bigcup_{i \in I} F(i)?
Back: The "index set".
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is \bigcup_{i \in I} F(i) alternatively denoted?
Back: \bigcup_{i \in I} F_i
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What kind of mathematical object is I in expression \bigcup_{i \in I} F(i)?
Back: A set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What kind of mathematical object is F in expression \bigcup_{i \in I} F(i)?
Back: A function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is \bigcup_{i \in I} F_i alternatively denoted?
Back: \bigcup_{i \in I} F(i)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What kind of mathematical object is F in expression \bigcup_{i \in I} F_i?
Back: A function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the domain of F assumed to be in expression \bigcup_{i \in I} F(i)?
Back: Some superset of I.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What condition must I satisfy in expression \bigcup_{i \in I} F(i)?
Back: N/A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose I = \{0, 1, 2\}. What does \bigcup_{i \in I} F(i) evaluate to?
Back: F(0) \cup F(1) \cup F(2)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose I = \varnothing. What does \bigcup_{i \in I} F(i) evaluate to?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What kind of mathematical object is F in expression \bigcap_{i \in I} F(i)?
Back: A function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is \bigcap_{i \in I} F(i) often alternatively denoted?
Back: \bigcap_{i \in I} F_i
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the domain of F assumed to be in expression \bigcap_{i \in I} F(i)?
Back: Some superset of I.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What condition must I satisfy in expression \bigcap_{i \in I} F(i)?
Back: I \neq \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose I = \{0, 1, 2\}. What does \bigcap_{i \in I} F(i) evaluate to?
Back: F(0) \cap F(1) \cap F(2)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose I = \varnothing. What does \bigcap_{i \in I} F(i) evaluate to?
Back: N/A. This is undefined.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is \bigcap_{i \in I} F_i alternatively denoted?
Back: \bigcap_{i \in I} F(i)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What kind of mathematical object is F in expression \bigcap_{i \in I} F_i?
Back: A function.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Function Sets
For sets A and B, the collection of functions F from A into B is: ^AB = {F \mid F \colon A \rightarrow B}
^AB is read as "B-pre-A". It is often written as B^A instead.
%%ANKI
Basic
For sets A and B, how is set B^A defined?
Back: \{F \mid F \colon A \rightarrow B\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For sets A and B, how is set ^AB defined?
Back: \{F \mid F \colon A \rightarrow B\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For any function F \colon A \rightarrow B, F is a subset of what other set?
Back: A \times B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For any function F \colon A \rightarrow B, F is a member of what other set?
Back: \mathscr{P}(A \times B)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For sets A and B, how is set B^A pronounced?
Back: As "B-pre-A".
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why prefer notation B^A over ^AB?
Back: The notation mirrors |B|^{|A|}, the number of elements in B^A given both sets are finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For sets A and B, how is set ^AB pronounced?
Back: As "B-pre-A".
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why prefer notation ^AB over B^A?
Back: Because the sets are written left-to-right, from domain to codomain.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why is set B^A denoted the way it is?
Back: If A and B are finite, then B^A has |B|^{|A|} members.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the domain of ^\omega\{0, 1\}?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the domain of a member of ^\omega\{0, 1\}?
Back: \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the range of \{0, 1\}^\omega?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the range of a member of \{0, 1\}^\omega?
Back: \{0, 1\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does \varnothing^\varnothing evaluate to?
Back: \{\varnothing\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For A \neq \varnothing, what does \varnothing^A evaluate to?
Back: \varnothing
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For A \neq \varnothing, why does \varnothing^A = \varnothing?
Back: No function can map a nonempty domain to an empty range.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For A \neq \varnothing, what does ^\varnothing A evaluate to?
Back: \{\varnothing\}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
For A \neq \varnothing, why does ^\varnothing A = \{\varnothing\}?
Back: \varnothing is the only function with empty domain.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).