Why do we tell functions from relations in structures?

Question

A relation is a set of ordered pairs (a,b)

A function is a relation (a,b) which satisfies the following conditions:

For all a, there is one and only one b

Therefore, all functions are relations.

In structures, we distinguish three kinds of symbols: functions, relations and constants

If all functions are relations, why don't we divide the symbols only into relations and constants?

Answer 1

It is true that every function symbol can be traded for a relation symbol along with some axioms stating that the symbol defines a function etc etc. You can also replace a constant symbol by a relation symbol and the axioms that there exists a unique element satisfying that relation.

But function and constant symbols make it easier to create new terms. And this allows for simplification in some aspects.

Consider the language of groups, with a binary operation $*$ and a constant $1$. If you replace them relations, then the axioms for a group will look like this:

1. $\exists y(1(y)\forall x(1(x)\rightarrow x=y))$ here we are saying that $1$ is a constant symbol.
2. $\forall x\forall y\exists z(*(x,y,z)\land\forall w(*(x,y,w)\rightarrow w=z))$ here we are saying that $*$ is a binary function symbol.
3. $\forall x\forall y\forall z(\exists w\exists u\exists v(*(x,y,w)\land *(w,z,u)\land *(y,z,v)\land *(x,v,u)))$ here we say that $*$ is associative.
4. $\forall x\exists y\exists z(*(x,y,z)\land 1(z))$ here we say that there exists an inverse for every element.
5. $\forall x\exists y(1(y)\land *(x,y,x)\land *(y,x,x))$ and here we say that $1$ is neutral for $*$.

Are these axioms really easier to handle compared to $\forall x\forall y\forall z((x*y)*z=x*(y*z))$? What about when you want to write an axiom stating that "Every element has order of at most $n$", then it will be some horribly long statement $\forall x\exists y_1\ldots\exists y_n(1(y_n)\land *(x,x,y_1)\land *(y_1,x,y_2)\land\ldots\land *(y_{n-1},x,y_n))$. How is this easier than $x*x*\ldots *x=1$, $n$ times?

So terms are useful, and we use functions and constant symbols for terms. This fact can be seem very clearly in set theory. The language only include the binary relations $\in,=$. But we add constants like $\varnothing$ and $\omega$ and so on; and we add functions like $\mathcal P(\bullet)$ for the power sets and $\bigcup\bullet$ for the union, and so on. We do it because it makes our lives so much easier.

But it is true, at the end of the day we can always go back to a relational language. Only that the cost is that we make every statement and proof vastly more complex, and introduce quantifiers everywhere. And this means that we also lose the ability to talk about "quantifier elimination" which is a useful property.