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More formally, a quotient graph is a quotient object in the category of graphs. The category of graphs is concretizable – mapping a graph to its set of vertices makes it a concrete category – so its objects can be regarded as "sets with additional structure", and a quotient graph corresponds to the graph induced on the quotient set V / R of ...
A 24-clue automorphic Sudoku with translational symmetry. Mathematics can be used to study Sudoku puzzles to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of clues in a valid puzzle?" and "In what ways can Sudoku grids be symmetric?" through the use of combinatorics and group theory.
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 ...
In mathematics, particularly in functional analysis, a seminorm is a norm that need not be positive definite.Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
Reflexivity: for every a, one has a = a.; Symmetry: for every a and b, if a = b, then b = a.; Transitivity: for every a, b, and c, if a = b and b = c, then a = c. [6] [7]Substitution: Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning.
The definitions for even and odd symmetry for complex-valued functions of a real argument are similar to the real case. In signal processing, a similar symmetry is sometimes considered, which involves complex conjugation. [4] [5] Conjugate symmetry:
When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra). The antisymmetry inherent in the exterior algebra means that when α ∧ β is viewed as a multilinear functional, it is alternating.
The symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a single equivalence class , the resulting quotient set can be properly metrized by the induced metric.