poochie_d said:But ordered tuples are really functions from [itex]\{1, ..., n\}[/itex] to (whatever set under consideration), right?
poochie_d said:But aren't tuples other than the ordered pair defined as functions, so that the definition of functions as triples would still be circular?
In set theory, a function is a relation between two sets, where each element in the first set (the domain) is paired with exactly one element in the second set (the codomain). This means that for every input, there is only one output.
In set theory, a function is defined as a subset of the Cartesian product of the domain and codomain, where each element in the domain is paired with exactly one element in the codomain. This can be represented using ordered pairs, with the first element being the input and the second element being the output.
In set theory, a function is a special type of relation where each input is paired with exactly one output. A relation, on the other hand, can pair multiple inputs with multiple outputs. In other words, a function is a subset of a relation that follows a specific rule.
A function can be represented in set notation as a set of ordered pairs, where the first element in each pair is an element from the domain and the second element is an element from the codomain. For example, the function f(x) = 2x could be written as {(1,2), (2,4), (3,6), ...}.
Functions are important in mathematics because they provide a way to map inputs to outputs, which is essential in solving many mathematical problems. Functions also allow us to describe and analyze relationships between sets in a precise and structured way.