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  2. Transitive relation - Wikipedia

    en.wikipedia.org/wiki/Transitive_relation

    In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x ...

  3. List of sums of reciprocals - Wikipedia

    en.wikipedia.org/wiki/List_of_sums_of_reciprocals

    The reciprocals of the non-negative integer powers of 2 sum to 2 . This is a particular case of the sum of the reciprocals of any geometric series where the first term and the common ratio are positive integers. If the first term is a and the common ratio is r then the sum is ⁠ r / a (r − 1) ⁠.

  4. 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".

  5. Additive inverse - Wikipedia

    en.wikipedia.org/wiki/Additive_inverse

    In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]

  6. 1 + 2 + 3 + 4 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E...

    Analytic continuation around the pole at s = 1 leads to a region of negative values, including ζ(−1) = ⁠− + 1 / 12 ⁠. In zeta function regularization, the series = is replaced by the series =. The latter series is an example of a Dirichlet series.

  7. Transitive closure - Wikipedia

    en.wikipedia.org/wiki/Transitive_closure

    A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitive relations include the equality relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if x ...

  8. Inverse function - Wikipedia

    en.wikipedia.org/wiki/Inverse_function

    If f(x)=y, then g(y)=x. The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with nonempty domain is injective if and only if it has a left inverse. [21] An elementary proof runs as follows: If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y ...

  9. Closure (mathematics) - Wikipedia

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

    [note 1] A subset X of S is said to be closed under these methods, if, when all input elements are in X, then all possible results are also in X. Sometimes, one may also say that X has the closure property. The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set.