Past exam of the logic and set theory course of the University of Oxford / 2009 / A / 3 / d / i

Replace each $n$-ary function symbol by an $(n+1)$-ary graph predicate, with axioms asserting exactly one output for each input tuple. Replace each constant by a unary predicate asserted to hold of exactly one element. Recursively introduce witnesses for term values, then translate formulas through their logical connectives and quantifiers as above.

Every original structure has a unique corresponding relational structure satisfying these axioms, on exactly the same carrier, and every such relational structure reconstructs the original functions and constants. Truth under assignments and logical consequences are preserved in both directions. In this precise relative sense, purely relational languages are adequate to represent first-order languages with functions and constants: expressive power is retained after adjoining the graph and uniqueness axioms. Arbitrary relational interpretations that fail those axioms are not claimed to represent functions.