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  2. Partition of a set - Wikipedia

    en.wikipedia.org/wiki/Partition_of_a_set

    Partitions of a 4-element set ordered by refinement. A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that ...

  3. Cartesian product - Wikipedia

    en.wikipedia.org/wiki/Cartesian_product

    A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be ...

  4. Product (mathematics) - Wikipedia

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

    In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) —where a ∈ A and b ∈ B. [5] The class of all things (of a given type) that have Cartesian products is called a Cartesian ...

  5. Product (category theory) - Wikipedia

    en.wikipedia.org/wiki/Product_(category_theory)

    In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

  6. Disjoint sets - Wikipedia

    en.wikipedia.org/wiki/Disjoint_sets

    Every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. [8] Disjoint-set data structures [9] and partition refinement [10] are two techniques in computer science for efficiently maintaining partitions of a set subject to ...

  7. Product topology - Wikipedia

    en.wikipedia.org/wiki/Product_topology

    In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology , which can also be given to a product space and which agrees ...

  8. Set (mathematics) - Wikipedia

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

    A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...

  9. Integer partition - Wikipedia

    en.wikipedia.org/wiki/Integer_partition

    The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part.