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# 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