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This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
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 ...
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".
An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called the incidence relation, specifying which points belong to which lines. An incidence structure can be specified by a family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to each line ...
1. The difference of two sets: x~y is the set of elements of x not in y. 2. An equivalence relation \ The difference of two sets: x\y is the set of elements of x not in y. − The difference of two sets: x−y is the set of elements of x not in y. ≈ Has the same cardinality as × A product of sets / A quotient of a set by an equivalence ...
One line indicates a single bond. Two lines indicate a double bond, and three lines indicate a triple bond. In some structures the atoms in between each bond are specified and shown. However, in some structures, the carbon molecules are not written out specifically. Instead, these carbons are indicated by a corner that forms when two lines connect.
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Other common double bonds are found in azo compounds (N=N), imines (C=N), and sulfoxides (S=O). In a skeletal formula, a double bond is drawn as two parallel lines (=) between the two connected atoms; typographically, the equals sign is used for this. [1] [2] Double bonds were introduced in chemical notation by Russian chemist Alexander Butlerov.