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sav08:predicate_logic_informally [2008/02/20 15:39] vkuncak |
sav08:predicate_logic_informally [2008/02/20 15:41] vkuncak |
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| In general all quantifiers range over some universal domain $D$. To restrict them to subsets of $D$, we can use bounded quantifiers: | In general all quantifiers range over some universal domain $D$. To restrict them to subsets of $D$, we can use bounded quantifiers: | ||
| - | * $\exists x \in S. P(x)$ means $\exists x. x \in S \and P(x)$ | + | * $\exists x \in S. P(x)$ means $\exists x. (x \in S \land P(x))$ |
| - | * $\forall x \in S. P(x)$ means $\forall x. x \in S \rightarrow P(x)$ (note implication instead of conjunction) | + | * $\forall x \in S. P(x)$ means $\forall x. (x \in S \rightarrow P(x))$ (note implication instead of conjunction) |
| More generally, if $\rho$ is some binary relation written in infix form, such as $<, \le, >, \ge$ we write | More generally, if $\rho$ is some binary relation written in infix form, such as $<, \le, >, \ge$ we write | ||
| - | * $\exists x \rho t. P(x)$ meaning $\exists x. x \rho t \and P(x)$ | + | * $\exists x \mathop{\rho} t. P(x)$ meaning $\exists x. (x \mathop{\rho} t \land P(x))$ |
| - | * $\forall x \rho t. P(x)$ means $\forall x. x \rho t \rightarrow P(x)$ (note implication instead of conjunction) | + | * $\forall x \mathop{\rho} t. P(x)$ means $\forall x. (x \mathop{\rho} t \rightarrow P(x))$ (note implication instead of conjunction) |
| ===== Evaluating First-Order Logic Formulas ===== | ===== Evaluating First-Order Logic Formulas ===== | ||