A ⊆ A. Euler and Venn diagrams An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If is a subset of , then the region representing is completely inside the region representing . If two sets have no elements in common, the regions do not overlap. A Venn diagram, in contrast, is a graphical representation of sets in which the loops divide the plane into zones such that for each way of selecting some of the sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are , , and , there should be a zone for the elements that are inside and and outside (even if such elements do not exist). Special sets of numbers in mathematics There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. Many of these important sets are represented in mathematical texts using bold (e.g. ) or blackboard bold (e.g. ) typeface. These include or , the set of all natural numbers: (often, authors exclude ); or , the set of all integers (whether positive, negative or zero): ; or , the set of all rational numbers (that is, the set of all proper and improper fractions): . For example, and ; or , the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as that cannot be rewritten as fractions, as well as transcendental numbers such as and ); or , the set of all complex numbers: , for example, . Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, represents the set of positive rational numbers. Functions A function (or mapping) from a set to a set is a rule that assigns to each "input" element of an "output" that is an element of ; more formally, a function is a special kind of relation, one that relates each element of to exactly one element of . A