Why Quantifier Free Formulas define Linear Functions.

Question

How do you prove that functions definable by quantifier-free formulas must be linear? I am interested for the structure $\langle Q,+,0\rangle$.

Answer 1

Such a function would be defined by setting it equal to a term in the above language. For example, your function definition might look something like $f(x, y, z) = 0 + x + x + y + z + 0 + z$. Using the axioms of the rational numbers, you can eliminate all $0$s and group variables together, so that the above definition would look like $f(x, y, z) = 2x + y + 2x$. (Note that "$2x$" is just shorthand for "$x + x$" and $2$ isn't actually in the language). This is clearly linear.

You can follow the same procedure for any function definable by a quantifier-free formula. I haven't been too rigorous but hopefully this answers your question.