A Relation from set A to set B is defined as a set of ordered pairs formed from the elements of set A and B.

In other words, a relation is a subset of the cartesian product of sets A and B. The subset is derived by establishing predicate filter(s) or criteria stating a condition that evaluates the qualifying ordered pairs from the cartesian product to be included in the subset as specified by the relation.

A relation is uni-directional i.e. if a relation exists from A to B then it does not imply that a relation exists from B to A as well.

Also, (a,b) ≠ (b,a)

Let a ∈ A and b ∈ B. Let (a,b) be an ordered pair of the relation R from set A to set B. ☞ Under this relation r, ‘b’ is called the image of ‘a’ and ‘a’ is called the preimage of ‘b’. ☞ Set A is called the Domain of the relation. ☞ Set B is called the Co-Domain of the relation. ☞ The set of all images is called the Range of the relation. ☞ The range of the relation is a subset of the Co-Domain. (Note that A and B are two different independent sets and the relation is applied on the independent elements of B and not the entire sets of B. This means that not all the elements of set B need to act as images of elements of A.) ☞ If A has m elements and B has n elements then their cartesian product will have m × n elements. The total number of relations possible will be the total number of subsets of the cartesian product A × B, i.e., 2mn

Finitary Relations

Relations can exist among many sets and not only two. For n number of sets S₁, S₂, S₃,…,S_n, a finitary relation is defined as a subset of the cartesian product S₁ × S₂× S₃ × … × S_n. Such relations are used to establish a connection between the members of the n-tuple.

Finitary Relations

Relations can exist among many sets and not only two. For n number of sets S₁, S₂, S₃,…,S_n, a finitary relation is defined as a subset of the cartesian product S₁ × S₂× S₃ × … × S_n. Such relations are used to establish a connection between the members of the n-tuple.

Degree of Relation

For finitary relations involving the ‘n’ number of sets, the relation is also termed as n-ary or n-adic relation. Such relations are also called relations of degree n. The finitary relation is also represented as R_s1…s2 or by _s1…s2R., where s_i ∈ S_i. This is read as “s₁, s₂, s₃, …, s_n, … s_n are R-related”. For Binary relations, we simply represent the relation as _xR_y.

I^th Domain of a Relation

In a relation involving S₁, S₂, S₃ ,…,S_n, the set of all elements of the S_i^th set which appear in the relation R is called the i^th domain of the relation (D_i). For binary relations, we have only D₁ and D₂. D₁ is called the domain and D₂ is called the range.

Homogeneous Relation

A special case of relation is when all the participating sets (S_i) are identical. Else, the set is called a Heterogeneous relation.

To illustrate an example of a finitary relation, consider an entrance examination in which students have to appear in three tests. Multiple attempts are allowed. Let us define a relation that constitutes the student’s name and the grades achieved in the three tests. The relation R is defined by:

The relation can also be illustrated in the form of a table as shown below. Such a relation is an example of a quaternary relation or a relation with degree 4.

In this example, each row represents the relation between a score of x in test 1, y in test 2 and z in test 3 on a given attempt.

Binary Relation

A binary relation is a finitary relation for which the degree is 2. A binary relation is defined in the context of two sets. For two sets A and B, a binary relation is a subset of the cartesian product of A and B (A × B). It is a set of ordered pairs (a,b) such that a ∈ A and b ∈ B. The degree of such relations is 2. The study of relations begins with binary relations as they are extensively studied. They are also simpler to work with. Binary Relations extends into the field of functions which are more important in the field of mathematics as they form the base for the most important aspect of mathematics – Calculus.