set theory
Set theory notations
Set Theory
Sets containing elements of various types, e.g. red, green, blue , | |||
The empty set (the set containing no elements) | |||
The element belongs to the set , e.g. , red red, green, blue | |||
The element does not belong to the set , e.g. , orange red, green, blue | |||
The cardinality or the number of elements belonging to the set , e.g. , red, green, blue , . | |||
The set is contained inside the set , thus all elements of are also elements of . In this situation we say that is a subset of . | |||
The sets and contain the same elements, e.g. a, b, cb, a, cc, b, a. The condition is true if and only if and . When does not contain the same elements as the notation is used. | |||
The set is strictly contained inside the set , in other words it is true that but . | |||
The set of elements satisfying the property , e.g. for it follows that and is even . | |||
The power set of is the set of all subsets of , i.e. . For example if then . | |||
The intersection of sets and , i.e. the set containing the elements that are found both in and . Thus and . For example . | |||
The intersection of sets , , ..., , i.e. the set containing the elements that are found in all of the given sets. In other words | |||
The union of sets and , i.e. the set containing the elements that are found in either or . Thus or . For example . | |||
The union of sets , , ..., , i.e. the set containing the elements that are found in at least one of the given sets. In other words | |||
The difference between and , i.e. the set containing the elements that are found in but are not found in . Thus and . For example . | |||
The Cartesian product of and , i.e. the set of all possible ordered pairs whose first component is an element of and whose second component is an element of . Thus and . For example . |
The set of natural numbers, | |||
The set of non-zero natural numbers, | |||
The set of integers, | |||
The set of non-zero integers, | |||
The set of non-positive integers, | |||
The set of negative integers, | |||
The set of rational numbers, | |||
The set of non-zero rational numbers, | |||
The set of non-positive rational numbers, | |||
The set of negative rational numbers, | |||
The set of non-negative rational numbers, | |||
The set of positive rational numbers, | |||
The set of real numbers, i.e. the union of the rationals and the irrationals | |||
The set of irrational numbers, | |||
The set of non-zero real numbers, | |||
The set of non-positive real numbers, | |||
The set of negative real numbers, | |||
The set of non-negative real numbers, | |||
The set of positive real numbers, | |||
Intervals on the real line defined through the sets: | |||
Intervals on the real line defined through the sets: | |||
Intervals on the real line defined through the sets: | |||
Intervals on the real line defined through the sets: | |||
The set of complex numbers, where is the imaginary unit satisfying | |||
The set of non-zero complex numbers, | |||
The -dimensional real coordinate space, where is a positive integer. This is basically the set containing all -tuples of real numbers, defined by |
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