Binary Relations

Definitions

Given a set of objects A, a binary relation R on the set is defined as a subset of A x A. The characteristic function r for R is the function r : A x A -> bool such that r(x, y) is true if (x, y) in R, and false if (x, y) not in R. For a more natural notation, we can use x ~ y to denote r(x, y).

More generally, a binary relation can be defined on a pair of sets A x B but to keep things simple, we’ll only cover binary relations over a single set.

Binary relations may have several properties. A few interesting ones are:

  • A binary relation is reflexive if for any x in A, x ~ x
  • A binary relation is strict or irreflexive if there is no x in A for which x ~ x
  • A binary relation is symmetric if for any x, y in A, x ~ y implies y ~ x
  • A binary relation is antisymmetric if for any x, y in A, x ~ y and y ~ x implies x = y
  • A binary relation is transitive if for any x, y, z in A, if x ~ y and y ~ z, then x ~ z
  • A binary relation is total if for any x, y in A, either x ~ y, y ~ x, or both (in other words, for any x, y, ~ imposes some relation between them)

Examples

The relation is in the subtree rooted at is a reflexive relation where A is the set of nodes of a tree. For any pair of nodes x and y, we can establish whether x is in the subtree rooted at y or not, and for any x, x ~ x is true.

The relation is parent of in a tree is a strict relation: for any x in the set of tree nodes A, x cannot be a parent of itself.

The relation edge between over the vertices of a non-directed graph is a symmetric relation: for any x and y vertices of the graph, if there is an edge from x to y, the same edge exists from y to x, in other words, if x ~ y then y ~ x.

The is in the subtree rooted at relation above is also antisymmetric: if for a pair of nodes we can say x is in the subtree rooted at y and also y is in the subtree rooted at x, it’s obvious that both x and y are, in fact, the root of the subtree, thus x ~ y.

The relation is reachable from over the vertices of a directed graph is a transitive relation: if x is reachable from y and y is reachable from z, then x is reachable from z.

All of the above examples are of total relations. An example of a non-total relation is is ancestor of in a tree. x can be an ancestor of y, in which case x ~ y, or y can be an ancestor of x, in which case y ~ x, but it could also be that x and y are in different subtrees, so neither x ~ y nor y ~ x holds.

Preorder

A preorder is a relation which is reflexive and transitive.

A preorder which is also symmetric is an equivalence. A preorder which is antisymmetric is a partial order. More on those below.

An example of preorder is the is reachable from relation over a directed graph in the example above. This relation is obviously reflexive and transitive, but it is neither symmetric nor antisymmetric. If x is reachable from y, it doesn’t mean that y is reachable from x, so symmetry is not guaranteed. Similarly, if x is reachable from y and y is reachable from x, it does not mean that y equals x.

Equivalence and Equality

An equivalence relation ~ is a binary relation that is reflexive, symmetric, and transitive. In other words, it is a preorder which also has the symmetric property.

Such a relation partitions the set over which it is defined into equivalence classes - groups of objects that are equivalent based on the relation.

An example of equivalence is same month over a set of dates. This relation is reflexive, since a date d has the same month as itself (d ~ d); is symmetric, since if d1 has the same month as d2, then d2 has the same month as d1 (d1 ~ d2 => d2 ~ d1); and transitive, since if d1 ~ d2 and d2 ~ d3 => d1 ~ d3.

This relation partitions our set of dates in the equivalence classes corresponding to dates in January, dates in February, and so on. Note that the dates for which the relation holds are equivalent, but not necessarily equal.

An equality relation is an equivalence relation which partitions the set A consisting of n objects into exactly n equivalence classes. In other words, for any x in A, only x ~ x is true.

Partial Order and Total Order

A partial order relation <= is a binary relation that is reflexive, antisymmetric, and transitive. In other words, it is a preorder which also has the antisymmetric property.

An example of a partial order is the is subset of relation. It is reflexive (A is a subset of A), antisymmetric (if A is a subset of B and B is a subset of A, then A = B), and transitive (if A is a subset of B and B is a subset of C, then A is a subset of C).

A total order relation is a partial order that is also total. The above example relation is subset of is not total - there could be a pair of sets A and B such that neither is the subset of the other.

An example of a total order relation is less than or equal to for integers.

Weak Order and Strict Weak Order

A weak order relation ~ is a binary relation that is transitive and total. This implies reflexivity (for any x and y, either x ~ y, y ~ x, or both, so for x and x we have x ~ x). In other words, it is a preorder which is also total.

An example of a weak order is less than or equal absolute value for complex numbers. For any two complex numbers c1 and c2, either c1 ~ c2 ( |c1| <= |c2|), c2 ~ c1 (|c2| <= |c1|), or both, so ~ is total. We also have c1 ~ c2 and c2 ~ c3 implies c1 ~ c3 (|c1| <= |c2| and |c2| <= |c3| implies |c1| <= |c3|). Unlike a total order though, the relation is not antisymmetric. We can have c1 ~ c2, c2 ~ c1, with c1 and c2 distinct complex numbers (any two numbers with the same absolute value but different components).

A strict weak order relation < is a binary relation that is transitive and strict (irreflexive).

An example of strict weak order is less than for integers.

Applications

Most programming languages provide a way to customize equality, inequality, and comparison operators (==, !=, <, <=, >, >=). There is an interesting point to be made about what equality means in this context. For some types, this can simply mean comparing the bits and if they are the same, the objects are equal. But we also have logical equality - two objects can have different bitwise values but still be considered equal. Even more so for comparing objects - comparing bit representations usually does not translate to a meaningful comparison of objects.

Note though that any other function bool r(const T& m1, const T& m2) or member function bool r(const T& other) of T denotes a binary relation on T.

Different algorithms require different types of relations to exist between objects.

For example, we need at least a partial order relation to perform a topological sort. That is, in an directed acyclic graph, we can sort the vertices such that for every edge from a to b, a precedes b in the order. This can be used, for example, on the dependency graph in a makefile to determine how to sequence work.

Having an equivalence relation (eg. ==), we can implement a linear search algorithm to traverse a data structure and find an object equivalent to a given object. The C++ standard library algorithm find is an example of such an algorithm.

Having a total order relation (eg. <=) or a strict weak order (eg. <), allows us to implement binary search over an ordered set of objects. A total order or strict weak order relation also enables comparison sort algorithms.

Similarly, we need a total oreder or strict weak order to be able to determine a minimum or a maximum element from a set of objects (min_element and max_element algorithms in C++).

Summary

  • A binary relation R on a set A is a subset of A x A, denoted by a characterisitc function r : A x A -> bool.
  • A binary relation on a type T is denoted by either a free function of the the form bool r(const T&, const T&) or a member function bool r(const T&).
  • A binary relations may have several properties: it can be reflexive or strict, symmetric or antisymmetric, transitive, total etc.
  • Depending on the properties it has, a relation can be, for example:
    • A preorder (reflexive and transitive)
    • An equivalence (reflexive, symmetric, and transitive)
    • A partial order (reflexive, antisymmetric, and transitive)
    • A weak order (reflexive, transitive, and total)
    • A strict weak order (irreflexive, transitive, and total)
  • Certain algorithms require the types they operate on to have relations with certain properties.