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Bijective function from N to the set E of even numbers. Although E is a proper subset of N, both sets have the same cardinality. N does not have the same cardinality as its power set P(N): For every function f from N to P(N), the set T = {n∈N: n∉f(n)} disagrees with every set in the range of f, hence f cannot be surjective.
Fundamentals. 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".
More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a one to one correspondence between the two sets n and S. The sets used to define natural numbers satisfy Peano axioms.
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. [1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero ( ) sets and it is by definition equal to the empty set.
Glossary of mathematical symbols. A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various ...
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 sets. [5] A set may have a finite number of elements or be an infinite set.
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [3] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the ...
Definition. The intersection of two sets and denoted by , [3] is the set of all objects that are members of both the sets and In symbols: That is, is an element of the intersection if and only if is both an element of and an element of [3] For example: The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.