18 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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Laws
The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.
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Cloze
{\cup} is to algebra of sets whereas {+} is to algebra of real numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cloze
{\cap} is to algebra of sets whereas {\cdot} is to algebra of real numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cloze
{-} is to algebra of sets whereas {-} is to algebra of real numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Cloze
{\subseteq} is to algebra of sets whereas {\leq} is to algebra of real numbers.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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%%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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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
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).
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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).
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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).
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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).
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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).
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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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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
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Basic
What kind of propositional logical statement are the antimonotonicity properties of \subseteq?
Back: An implication.
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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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).
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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).
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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).
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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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Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).