Help with first order peanos arithemtic

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What does A(c,d) represent X mean?

There are few definitions in mathematical logic that don't vary between textbooks. I see two options.

(1) A represents X if $\mathcal{N}\models A(\vec{x})$ iff $\vec{x}\in X$ for all $\vec{x}\in\mathbb{N}^2$ .

(2) A represents X in PA if
(a) $\mathrm{PA}\vdash A(\vec{x})$ for all $\vec{x}\in X$ , and
(b) $\mathrm{PA}\vdash \neg A(\vec{x})$ for all $\vec{x}\notin X$ .

Here $\mathcal{N}$ is the standard model of natural numbers, PA is Peano arithmetic, and if $\vec{x}$ is a pair of natural numbers, then $A(\vec{x})$ denotes A with corresponding numerals substituted for the variables c, d.

Basically, representability can be relative to truth in an interpretation or relative to provability in a theory. Representable functions are usually defined relative to provability in PA. See "Computability and logic" by Boolos and Jeffrey, chapter "Representability of Recursive Functions."

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also why does (m,n) element X iff A(m,n) interpreted in the natural way as a statement in number theory is true?

Under option (1) above this is true by definition.