Set (mathematics)

A race is a mathematical model for the collection of different things; a vintage contains elements or members, which can be mathematical objects of all kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no component is the empty set; a set with a single component is a singleton. A set may cause a finite number of elements or be an infinite set. Two sets are represent if they realise precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the specifics way to supply rigorous foundations for all branches of mathematics since the first half of the 20th century.

Principle of inclusion as well as exclusion

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set & the size of their intersection are known. It can be expressed symbolically as | A B | = | A | + | B | | A B | . {\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}

A more general form of the principle can be used to find the cardinality of any finite union of sets: | A 1 A 2 A 3 A n | = | A 1 | + | A 2 | + | A 3 | + | A n | | A 1 A 2 | + | A 1 A 3 | + | A n 1 A n | + + 1 n 1 | A 1 A 2 A 3 A n | . {\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right\\&{}-\left\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right\\&{}+\ldots \\&{}+\left-1\right^{n-1}\left\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right.\end{aligned}}}