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).
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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).
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).
<!--ID: 1717554445662-->
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).
<!--ID: 1717554445665-->
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).
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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).
<!--ID: 1717679397781-->
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).
<!--ID: 1717679397797-->
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).
<!--ID: 1717679397825-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1717679397836-->
END%%
%%ANKI
Basic
Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
Back: $\mathscr{P}\mathscr{P}A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881694-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881698-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881702-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881705-->
END%%
%%ANKI
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881712-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
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).
<!--ID: 1716803270441-->
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).
<!--ID: 1716803270447-->
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).
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).
<!--ID: 1717372322271-->
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).
<!--ID: 1717372322264-->
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).
<!--ID: 1717372322275-->
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).
<!--ID: 1717366846568-->
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).
<!--ID: 1717366846580-->
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).
<!--ID: 1717366846573-->
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).
<!--ID: 1717366846594-->
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).
<!--ID: 1717367767303-->
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).
<!--ID: 1717367767308-->
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).
<!--ID: 1717367767311-->
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).
<!--ID: 1717372322278-->
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).
<!--ID: 1717372322281-->
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).
<!--ID: 1717372322284-->
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).
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 algebra of sets operators is 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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881718-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881723-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881726-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881731-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881735-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881742-->
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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1718069881752-->
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).
<!--ID: 1718069881759-->
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).
<!--ID: 1718069881766-->
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).
<!--ID: 1718069881773-->
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).
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).
<!--ID: 1717367767316-->
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).
<!--ID: 1717367767323-->
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).
<!--ID: 1717367767320-->
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).
<!--ID: 1717373048517-->
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).
<!--ID: 1717373048522-->
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).
<!--ID: 1717373048525-->
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).
<!--ID: 1717367767328-->
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).
<!--ID: 1717368301050-->
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).
<!--ID: 1717368301055-->
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).
<!--ID: 1717372322295-->
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).
<!--ID: 1717372322299-->
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).
<!--ID: 1717372322304-->
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
What kind of propositional logical statement are the monotonicity properties of $\subseteq$?
Back: An implication.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717073536967-->
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).
<!--ID: 1717073536973-->
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).
<!--ID: 1717073536976-->
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).
<!--ID: 1717073536979-->
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).
<!--ID: 1717073536982-->
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).
* $A \subseteq B \Rightarrow A \times C \subseteq B \times C$
%%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](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
