Why all functions are relations but not all relations are functions?

Answer 1

In mathematics, a relation is a set of ordered pairs, where each input is related to one or more outputs. A function is a specific type of relation where each input is related to exactly one output. Therefore, all functions are relations because they involve a set of ordered pairs, but not all relations are functions because some relations may have an input related to multiple outputs, violating the definition of a function.

Answer 2

I'm pretty sure that no one has actually wrote a complete proof of this statement, but the concept centers around the fact that functions are relations where the codomain is dependent on the domain, but relations don't necessarily have to be. If that sounds vague, it's because no one has really came up with a precise answer yet.

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Actually, the definition of a function ensures that it is a relation. Given that, all you need to do is to find one relation that is not a function. A popular one is y = sqrt(x) for x ≥ 0. Since each value of x (other than 0), is mapped onto 2 distinct values of y the relation is not a function. However, it is easily made into function by limiting the codomain to non-negative reals or to non-positive reals.

Similarly, relations such as reciprocal or logarithm can be made into functions by defining the domain or codomain to get around the exceptions.

Answer 3

Yes