41 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.
%%ANKI
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).
END%%
%%ANKI
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).
END%%
Symmetric Difference
Define the symmetric difference of sets A
and B
as $A \mathop{\triangle} B = (A - B) \cup (B - A)
$
%%ANKI
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).
END%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI Basic Is the symmetric difference commutative? Back: Yes. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic Is the symmetric difference associative? Back: Yes. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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\}
$
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%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).
END%%
%%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.
END%%
%%ANKI
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%%
%%ANKI
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).
END%%
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)}
%%ANKI
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).
END%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
Basic
Assume AoC and H(j) \neq \varnothing
for all j \in I
. What does \bigtimes_{i \in I} H(i)
evaluate to?
Back: \{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%%
%%ANKI
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).
END%%
%%ANKI
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*}
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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.
END%%
%%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.
END%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
Basic
Under what three conditions is A \times B = B \times A
?
Back: A = B
or A = \varnothing
or B = \varnothing
.
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*}
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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*}
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
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*}
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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*}
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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*}
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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*}
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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*}
%%ANKI
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%%
%%ANKI
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 the following? $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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
In addition,
A \subseteq B \Rightarrow A \times C \subseteq B \times C
%%ANKI
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.
END%%
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%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).
END%%
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).
END%%
%%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).
END%%
Index Sets
Let I
be a set, called the index set. Let F
be a set/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 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).