What is the Difference Between Relation and Function?
🆚 Go to Comparative Table 🆚The difference between a relation and a function lies in the number of outputs associated with each input.
- Relation: A relation is a set of ordered pairs featuring an object from one set (let's call it set X) and an object from another set (let's call it set Y). In a relation, one input can be associated with multiple outputs, forming a many-to-one relationship. For example, consider the set {(2, x), (9, y), (2, z)}. This is not a function because the input "2" is associated with both "x" and "z".
- Function: A function is a specific type of relation where each input in set X has exactly one output in set Y. In other words, a function maps each element in set X to a single element in set Y. For example, consider the set {(2, x), (9, y), (5, x)}. This is a function because each input in set X has a unique output in set Y.
To summarize, a relation is a set of ordered pairs with elements from two sets, while a function is a specific type of relation that maps each input to a single output. All functions are relations, but not all relations are functions. The key differentiating factor between a relation and a function is that a function has a single input for a single output, whereas a relation can have many outputs for a single input.
Comparative Table: Relation vs Function
The main difference between a relation and a function is that a relation can have multiple outputs for a single input, while a function has a single input for a single output. Here is a table summarizing the differences between relations and functions:
| Differentiating Parameter | Relations | Functions |
|---|---|---|
| Definition | A relation is a collection of ordered pairs, showing a relationship between two sets. | A function is a relation with only one output for each input, and it can be represented using the function notation (e.g., F(x) or f(x)). |
| Denotation | A relation is denoted by "R" or by the symbol "r". | A function is denoted by "F" or "f". |
| Example | R = {(2, x), (9, y), (2, z)} - This is not a function because "2" is an input for both x and z. | F = {(2, x), (9, y), (5, x)} - This is a function because each input in set X has exactly one output in set Y. |
| Note | Every relation is not a function. | Every function is a relation, but not every relation is a function. |
Both relations and functions can be represented in similar ways, such as through tables, graphs, and mappings. However, the key difference lies in the number of outputs for each input: a relation can have multiple outputs for a single input, while a function has a single output for each input.
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