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Relation Theory Terms And Formulas Class Free And Bound Variables

1	Class
1	Terms And Formulas
1	Relation Theory
2	Free And Bound Variables

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Universal Formula

If $\Phi$ is a class of -ary relations with $\vec{x}$ as the only free variables, an -ary formula $\psi$ is universal for $\Phi$ if for any $\phi\in\Phi$ there is some such that $\psi(e,\vec{x})\leftrightarrow\phi(\vec{x})$ . In other words, $\psi$ can simulate any element of $\Phi$ .

Similarly, if $\Phi$ is a class of function of $\vec{x}$ , a formula $\psi$ is universal for $\Phi$ if for any $\phi\in\Phi$ there is some such that $\psi(e,\vec{x})=\phi(\vec{x})$ .