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Function**

Given sets \(X,Y\), a function \(f:X\to Y\) is an assignment of precisely one element \(f(x)\) of \(Y\) to each element \(x\) of \(X\), written \(x\mapsto f(x)\). This is equivalent to there being a functional relation \(\mathscr{R}\) on \(X\) such that the image lies within \(Y\); \(\forall x\in X:\exists!y\in Y:x\mathscr{R}y\), with \(y=f(x)\Leftrightarrow x\mathscr{R}y\).