Solutions to Chapter 27 Sentences of FOL

A. Identify which variables are bound and which are free. We underline the bound variables, and overline the free variables.

1. 1.
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   $\exists xL(\underset{¯}{x},\overline{y})\wedge \forall yL(\underset{¯}{y},\overline{x})$

2. 2.
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   $\forall xA(\underset{¯}{x})\wedge B(\overline{x})$

3. 3.
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   $\forall x(A(\underset{¯}{x})\wedge B(\underset{¯}{x}))\wedge \forall y(C(\overline{x})\wedge D(\underset{¯}{y}))$

4. 4.
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   $\forall x\exists y[R(\underset{¯}{x},\underset{¯}{y})\to (J(\overline{z})\wedge K(\underset{¯}{x}))]\vee R(\overline{y},\overline{x})$

5. 5.
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   $\forall x_1(M(\overline{x_2})\leftrightarrow L(\overline{x_2},\underset{¯}{x_1}))\wedge \exists x_{2}L(\overline{x_3},\underset{¯}{x_2})$