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3.1 Formulas for binary set operations ⋂, ... 3.5.1.1 Empty set. 3.5.2 Meets, Joins, ... Sets that do not intersect are said to be disjoint.
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.
The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A. It is the smallest convex set containing A . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set.
Russell's paradox concerns the impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither contain itself (because its members all do not contain themselves) nor avoid containing itself (because if it did, it should be included as one of its members). [2]
Precisely, a binary relation over sets and is a set of ordered pairs (,) where is in and is in . [2] It encodes the common concept of relation: an element x {\displaystyle x} is related to an element y {\displaystyle y} , if and only if the pair ( x , y ) {\displaystyle (x,y)} belongs to the set of ordered pairs that defines the binary relation.
Symmetric difference of sets A and B, denoted A B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1, 2, 3} and {2, 3, 4}, the symmetric difference set is {1, 4}.
Cantor's diagonal argument (among various similar names [note 1]) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers.
map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function f and a collection of elements, and as the result, returns a new collection with f applied to each element from the collection.