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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|MATH202 EXPRESSIVE COMPLETENESS

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|EXPRESSIVE COMPLETENESS

This and the next section study the relation between formulae of $\mathrm{PL}$ and boolean functions established by Definition 5.7, according to which every $\mathrm{PL}$ formula defines a boolean function. The question now is the opposite: Can every boolean function be defined by some formula of PL?

Introducing abbreviations $\wedge, \vee$ and others in Section 5.1.1, we remarked that they are not necessary but merely convenient. Their being “not necessary” means that any function which can be defined by a formula containing these connectives, can also be defined by a formula which does not contain them. E.g., a function defined using $\vee$ can be also defined using $\neg$ and $\rightarrow$.
Concerning our main question we need a stronger notion of an expressively complete set of connectives, namely, one allowing to define all boolean functions.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|DISJUNCTIVE AND CONJUNCTIVE NORMAL FORMS

The fact that, for instance, ${\neg, \rightarrow}$ is an expressively complete set, vastly reduces the need for elaborate syntax when studying propositional logic. We can (as we indeed have done) restrict the syntax of WFF ${ }^{P L}$ to the necessary minimum. This simplifies many proofs concerned with the syntax and the axiomatic systems since such proofs involve often induction on the syntactic definitions (of WFF, of $\vdash$, etc.). Expressive completeness of a set of connectives means that any entity (any function defined by a formula) has some specific, “normal” form using only the connectives from the set.
Now we will show that even more “normalization” is possible. Not only every boolean function can be defined by some formula using only the connectives from one expressively complete set – every such a function can be defined by such a formula which, in addition, has a very specific form.
Definition 6.5 A formula $B$ is in
(1) disjunctive normal form, DNF, iff $B=C_1 \vee \ldots \vee C_n$, where each $C_i$ is a conjunction of literals.
(2) conjunctive normal form, CNF, iff $B=D_1 \wedge \ldots \wedge D_n$, where each $D_i$ is a disjunction of literals.

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## 数学代写|数理逻辑入门代写数学逻辑导论代考|析取和合取的正常形式

(1)析取正常形式，DNF, iff $B=C_1 \vee \ldots \vee C_n$，其中每个$C_i$是一个字面值的连接
(2)析取正常形式，CNF, iff $B=D_1 \wedge \ldots \wedge D_n$，其中每个$D_i$是一个字面值的析取

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