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- Simple_theorems_in_the_algebra_of_sets abstract "The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix ∪), intersection (infix ∩), and set complement (postfix ') of sets.These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}. The algebra of sets describes the properties of all possible subsets of U, called the power set of U and denoted P(U). P(U) is assumed closed under union, intersection, and set complement. The algebra of sets is an interpretation or model of Boolean algebra, with union, intersection, set complement, U, and {} interpreting Boolean sum, product, complement, 1, and 0, respectively.The properties below are stated without proof, but can be derived from a small number of properties taken as axioms. A \"*\" follows the algebra of sets interpretation of Huntington's (1904) classic postulate set for Boolean algebra. These properties can be visualized with Venn diagrams. They also follow from the fact that P(U) is a Boolean lattice. The properties followed by \"L\" interpret the lattice axioms.Elementary discrete mathematics courses sometimes leave students with the impression that the subject matter of set theory is no more than these properties. For more about elementary set theory, see set, set theory, algebra of sets, and naive set theory. For an introduction to set theory at a higher level, see also axiomatic set theory, cardinal number, ordinal number, Cantor–Bernstein–Schroeder theorem, Cantor's diagonal argument, Cantor's first uncountability proof, Cantor's theorem, well-ordering theorem, axiom of choice, and Zorn's lemma.The properties below include a defined binary operation, relative complement, denoted by infix \"\\\". The \"relative complement of A in B,\" denoted B \\A, is defined as (A ∪B′)′ and as A′ ∩B.PROPOSITION 1. For any U and any subset A of U:{}′ = U;U′ = {};A \\ {} = A;{} \\ A = {};A ∩ {} = {};A ∪ {} = A; *A ∩ U = A; *A ∪ U = U;A′ ∪ A = U; *A′ ∩ A = {}; * A \\ A = {};U \\ A = A′;A \\ U = {};A′′ = A;A ∩ A = A;A ∪ A = A.PROPOSITION 2. For any sets A, B, and C:A ∩ B = B ∩ A; * LA ∪ B = B ∪ A; * LA ∪ (A ∩ B) = A; LA ∩ (A ∪ B) = A; L(A ∪ B) \\ A = B \\ A;A ∩ B = {} if and only if B \\ A = B;(A′ ∪ B)′ ∪ (A′ ∪ B′)′ = A;(A ∩ B) ∩ C = A ∩ (B ∩ C); L(A ∪ B) ∪ C = A ∪ (B ∪ C); LC \\ (A ∩ B) = (C \\ A) ∪ (C \\ B);C \\ (A ∪ B) = (C \\ A) ∩ (C \\ B);C \\ (B \\ A) = (C \\ B) ∪(C ∩ A);(B \\ A) ∩ C = (B ∩ C) \\ A = B ∩ (C \\ A);(B \\ A) ∪ C = (B ∪ C) \\ (A \\ C).The distributive laws: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); * A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). *PROPOSITION 3. Some properties of ⊆:A ⊆ B if and only if A ∩ B = A;A ⊆ B if and only if A ∪ B = B;A ⊆ B if and only if A′ ∪ B;A ⊆ B if and only if B′ ⊆ A′;A ⊆ B if and only if A \\ B = {};A ∩ B ⊆ A ⊆ A ∪ B.".
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- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Algebra_of_sets.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Algebraic_structure.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Axiom.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Axiom_of_choice.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Boolean_algebra.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Boolean_algebra_(structure).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Cantors_diagonal_argument.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Cantors_theorem.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Cardinal_number.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Category:Basic_concepts_in_set_theory.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Closure_(mathematics).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Complement_(set_theory).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Discrete_mathematics.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Distributive_property.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Edward_Vermilye_Huntington.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Empty_set.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Georg_Cantors_first_set_theory_article.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink If_and_only_if.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Infix.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Interpretation_(logic).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Intersection_(set_theory).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Lattice_(order).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Logical_conjunction.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Logical_disjunction.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Mathematical_proof.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Model_theory.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Naive_set_theory.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Negation.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Ordinal_number.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Postfix.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Power_set.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Schröder–Bernstein_theorem.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Set_(mathematics).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Set_theory.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Subset.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Union_(set_theory).
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Universal_set.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Venn_diagram.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Well-ordering_theorem.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLink Zorns_lemma.
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLinkText "Simple theorems in the algebra of sets".
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLinkText "Simple_theorems_in_the_algebra_of_sets".
- Simple_theorems_in_the_algebra_of_sets wikiPageWikiLinkText "simple theorems in the algebra of sets".
- Simple_theorems_in_the_algebra_of_sets subject Category:Basic_concepts_in_set_theory.
- Simple_theorems_in_the_algebra_of_sets type Concept.
- Simple_theorems_in_the_algebra_of_sets comment "The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix ∪), intersection (infix ∩), and set complement (postfix ') of sets.These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}. The algebra of sets describes the properties of all possible subsets of U, called the power set of U and denoted P(U). P(U) is assumed closed under union, intersection, and set complement.".
- Simple_theorems_in_the_algebra_of_sets label "Simple theorems in the algebra of sets".
- Simple_theorems_in_the_algebra_of_sets sameAs Q7520837.
- Simple_theorems_in_the_algebra_of_sets sameAs Q7520837.
- Simple_theorems_in_the_algebra_of_sets wasDerivedFrom Simple_theorems_in_the_algebra_of_sets?oldid=528695478.
- Simple_theorems_in_the_algebra_of_sets isPrimaryTopicOf Simple_theorems_in_the_algebra_of_sets.