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  2. Transitive relation - Wikipedia

    en.wikipedia.org/wiki/Transitive_relation

    The transitive extension of R 1 would be denoted by R 2, and continuing in this way, in general, the transitive extension of R i would be R i + 1. The transitive closure of R, denoted by R* or R ∞ is the set union of R, R 1, R 2, ... . [8] The transitive closure of a relation is a transitive relation. [8]

  3. Proofs involving the addition of natural numbers - Wikipedia

    en.wikipedia.org/wiki/Proofs_involving_the...

    The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.

  4. Euclidean geometry - Wikipedia

    en.wikipedia.org/wiki/Euclidean_geometry

    Things that are equal to the same thing are also equal to one another (the transitive property of a Euclidean relation). If equals are added to equals, then the wholes are equal (Addition property of equality). If equals are subtracted from equals, then the differences are equal (subtraction property of equality).

  5. Closure (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Closure_(mathematics)

    Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the smallest equivalence relation that contains it. Other examples [ edit ]

  6. Algebra of sets - Wikipedia

    en.wikipedia.org/wiki/Algebra_of_sets

    The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".

  7. Commutative property - Wikipedia

    en.wikipedia.org/wiki/Commutative_property

    The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...

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  9. Ring (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Ring_(mathematics)

    The Burnside ring's additive group is the free abelian group whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents.

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