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title TARGET DECK FILE TAGS tags
Algebra of Sets Obsidian::STEM algebra::set set
algebra
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%%

Laws

The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

%%ANKI 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).

END%%

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%%

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%%

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%%

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%%

Bibliography

  • Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).