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In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] 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 ...
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is reflexive), then the qualification proper requires the objects to be different. For example, a proper subset of a set S is a subset of S that is different from S, and a proper divisor of a number n is a divisor of n that is different from n.
Mathematical relations fall into various types according to their specific properties, often as expressed in the axioms or definitions that they satisfy. Many of these types of relations are listed below.
In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x ...
For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal." ∵ Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ∵ it has no positive integer factors other than itself and one." ∋ 1. Abbreviation of "such that".
For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X {\displaystyle X} and Y {\displaystyle Y} are sets and L ⊆ X × Y {\displaystyle L\subseteq X\times Y} is a relation from X {\displaystyle X} to Y , {\displaystyle Y,} then L T {\displaystyle L^{\operatorname {T} }} is the relation defined ...
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For example, the defining condition of a zigzag poset is written as a 1 < a 2 > a 3 < a 4 > a 5 < a 6 > ... . Mixed chained notation is used more often with compatible ...