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数学代考| Frege 离散数学代写

数学代写| Frege 离散代考

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离散数学代写

based on degrees of truth, rather than on the standard binary truth-values of ‘true or false’ (1 or 0) of propositional logic. That is, while statements made in propositional logic are either true or false ( 1 or 0$)$, the truth-value of a statement made in fuzzy logic is a value between 0 and 1. Its value expresses the extent to which the statement is true, with a value of 1 expressing absolute truth, and a value of 0 expressing absolute falsity.

Fuzzy logic uses degrees of truth as a mathematical model for vagueness, and this is useful since statements made in natural language are often vague and have a certain (rather than an absolute) degree of truth. It is an extension of classical logic to handle the concept of partial truth, where the truth-value lies between completely true and completely false. Lofti Zadeh developed fuzzy logic at Berkeley in the $1960 \mathrm{~s}$, and it has been successfully applied to Expert Systems and other areas of Artificial Intelligence.

For example, consider the statement “John is tall”. If John were 6 foot, 4 inches, then we would say that this is a true statement (with a truth value of 1) since John is well above average height. However, if John is 5 feet, 9 inches tall (around average height), then this statement has a degree of truth, and this could be indicated by a fuzzy truth valued of $0.6$. Similarly, the statement that today is sunny may be assigned a truth-value of 1 if there are no clouds, $0.8$ if there are a small number of clouds and 0 if it is raining all day.

Propositions in fuzzy logic may be combined together to form compound propositions. Suppose $X$ and $Y$ are propositions in fuzzy logic, then compound propositions may be formed from the conjunction, disjunction and implication operators. The usual definition in fuzzy logic of the truth-values of the compound propositions formed from $X$ and $Y$ is given by
\begin{aligned} &\text { Truth }(\neg X)=1 \text {-Truth }(X), \ &(X \text { and } Y)=\min (\text { Truth }(X), \operatorname{Truth}(Y)), \end{aligned}
Truth $(X$ and $Y)=\min ($ Truth $(X)$, Truth $(Y))$, Truth $(X$ or $Y)=\max (\operatorname{Truth}(X)$, Truth $(Y))$,
Truth $(X \rightarrow Y)=\operatorname{Truth}(\neg X$ or $Y)$
There is another way in which the operators may be defined in terms of multiplication:
$$\text { Truth }(X \text { and } Y)=\operatorname{Truth}(X) * \operatorname{Truth}(Y),$$
Truth $(X$ or $Y)=1-(1-\operatorname{Truth}(X)) *(1-\operatorname{Truth}(Y))$
Truth $(X \rightarrow Y)=\max {z \mid \operatorname{Truth}(X) * z \leq \operatorname{Truth}(Y)}$ where $0 \leq z \leq 1$
$16.2$ Fuzzy Logic
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Under these definitions, fuzzy logic is an extension of classical two-valued logic, which preserves the usual meaning of the logical connectives of propositional logic when the fuzzy values are just ${0,1}$.

Fuzzy logic has been useful in the expert system and artificial intelligence applications, and it has also been applied to the aerospace and automotive sectors, and to the medical, robotics and transport sectors. The first fuzzy logic controller was developed in England, in the mid-1970s.

图论代考

$$\开始{对齐} &\text { 真相 }(\neg X)=1 \text {-真相 }(X), \ &(X \text { and } Y)=\min (\text { Truth }(X), \operatorname{Truth}(Y)), \end{对齐}$$

$$\text { 真相 }(X \text { 和 } Y)=\operatorname{真相}(X) * \operatorname{真相}(Y),$$

$16.2$ 模糊逻辑
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密码学代考

• Cryptosystem
• A system that describes how to encrypt or decrypt messages
• Plaintext
• Message in its original form
• Ciphertext
• Message in its encrypted form
• Cryptographer
• Invents encryption algorithms
• Cryptanalyst
• Breaks encryption algorithms or implementations

编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码