Relation vs Function
From high school mathematics onwards, function becomes a common term. Even though it is used quite often, it is used without proper understanding of its definition and interpretations. This article focuses on describing those aspects of a function.
A relation is a link between the elements of two sets. In a more formal setting, it can be described as a subset of the Cartesian product of two sets X and Y. Cartesian Product of X and Y, denoted as X×Y, is a set of ordered pairs consisting of elements from the two sets, often denoted as (x,y). The sets do not have to be different. For example, a subset of elements from A×A, is called a relation on A.
Functions are a special type of relations. This special type of relation describes how one element is mapped to another element in another set or the same set. For the relation to be a function, two specific requirements have to be satisfied.
Every element of the set where each mapping starts must have an associated/linked element in the other set.
The elements in the set where mapping starts can only be associated/linked to one and only one element in the other set
The set from which the relation is mapped is known as the Domain. The set, where the relation is mapped into is known as the Codomain. The subset of elements in the codomain containing only the elements linked to the relation is known as the Range.
Technically, a function is a relation between two sets, where each element in one set is uniquely mapped to an element in the other.
Notice the following
- Every element in the domain is mapped into the codomain.
- Several elements of the domain are connected to the same value in the codomain, but a single element from the domain cannot be connected to more than one element of the codomain. (Mapping has to be unique)
- If every single element of the domain is mapped into distinct and unique elements in the codomain, the function is said to be a “one-to-one” function.
- Codomain contains element other than the ones connected to the elements of the domain. The range does not have to be the codomain. If the codomain is equal to the range, the function is known as an “onto” function.
When the values that can be taken by the function are real, it is called a real function. The elements of codomain and domain are real numbers.
Functions are always denoted using variables. The elements of the codomain are symbolically represented by the variable. The notation f(x) represents the elements of the range. The relation can be represented using the expression in the form f(x)=x^2. It says that the element of the domain is mapped into the square of the element, within the codomain.
What is the difference between Function and Relation?
• Functions are a special type of relations.
• Relation is based on the Cartesian product of two sets.
• Function is based on relations with specific properties.
• Domain of a function has to be mapped into the codomain such that each element has a uniquely determined, corresponding value in the codomain. Relation can link single element to multiple values.