In mathematics, the terms ‘relation’ and ‘function’ are used to describe relationships between sets of numbers or objects, but they have distinct meanings.
A relation is a connection or association between two sets. Formally, a relation is a set of ordered pairs, where each pair consists of an input from one set and an output from another set. For example, if we have a set of inputs, say {1, 2, 3}, and a set of outputs, say {4, 5, 6}, we can create a relation such as {(1, 4), (2, 5), (3, 6)}. Here, each input is paired with an output, but there are no restrictions on how many outputs can be associated with each input.
On the other hand, a function is a special type of relation. A function must satisfy the rule that each input is related to exactly one output. Using the previous example, if we have a set of inputs {1, 2, 3} and outputs {4, 5}, a valid function could be {(1, 4), (2, 5)}. However, if we tried to include the pair (1, 5), it would no longer be a function because the input ‘1’ is now associated with two different outputs (4 and 5).
In summary, all functions are relations, but not all relations are functions. The key distinction is that a function has a specific rule that each input relates to only one output, while a relation does not have this restriction.