数学代写代考|Strong Induction离散数学代写 - 代考代写：100%准时可靠您的作业代写专家

Strong induction is another form of mathematical induction, which is often employed when we cannot prove a result with (weak) mathematical induction. It is similar to weak induction in that there is a base step and an inductive step. The base step is identical to weak mathematical induction, and it involves showing that the statement is true for some natural number (usually the number 1). The inductive step is a little different, and it involves showing that if the statement is true for all the natural numbers less than or equal to an arbitrary number $k$, then the statement is true for its successor $k+1$. This is often written as $(P(1) \wedge P(2) \wedge \ldots \wedge$ $P(k)) \rightarrow P(k+1) .$
4.2 Strong Induction
83
From the base step and the inductive step, we infer that the statement is true for all the natural numbers (that are greater than or equal to the number specified in the base case). Formally, the proof technique used in mathematical induction is of the form $^{2}$ :
$$
(P(1) \wedge \forall k[(P(1) \wedge P(2) \wedge \ldots \wedge P(k)) \rightarrow P(k+1)]) \rightarrow \forall n P(n)
$$
Strong and weak mathematical induction are equivalent in that any proof done by weak mathematical induction may also be considered a proof using strong induction, and a proof conducted with strong induction may also be converted into a proof using weak induction.

Weak mathematical induction is generally employed when it is reasonably clear how to prove $P(k+1)$ from $P(k)$, with strong mathematical typically employed where it is not so obvious. The validity of both forms of mathematical induction follows from the well-ordering property of the Natural Numbers, which states that every non-empty set has a least element.
Well-Ordering Principle
Every non-empty set of natural numbers has a least element. The well-ordering principle is equivalent to the principle of mathematical induction.

Example 4.3 Show that every natural number greater than one is dividie by prime number. Proof Base Case and is divisible by itself. and is divisible by itself.
Inductive Step (strong induction)
Suppose that the result is true for every number less than or equal to $k$. Then we consider $k+1$, and there are two cases to consider. If $k+1$ is prime, then it is divisible by itself. Otherwise, it is composite and it may be factored as the product of two numbers each of which is less than or equal to $k$. Each of these numbers is divisible by a prime number by the strong inductive hypothesis, and so $k+1$ is divisible by a prime number.

Thus, we have shown that if all the natural numbers less than or equal to $k$ are divisible by a prime number, then $k+1$ is divisible by a prime number. We have shown that the base case $\mathrm{P}(2)$ is true, and so it follows from strong mathematical induction that every natural number greater than one is divisible by some prime number.

图论代考

强归纳是数学归纳的另一种形式，当我们无法用（弱）数学归纳证明结果时，通常会使用这种形式。它与弱归纳类似，有一个基本步骤和一个归纳步骤。基本步骤与弱数学归纳法相同，它涉及证明该陈述对于某个自然数（通常是数字 1）为真。归纳步骤稍有不同，它涉及证明如果该陈述对小于或等于任意数 $k$ 的所有自然数为真，则该陈述对其后继 $k+1$ 为真。这通常写成 $(P(1) \wedge P(2) \wedge \ldots \wedge$ $P(k)) \rightarrow P(k+1) .$
4.2 强感应
83
从基本步骤和归纳步骤，我们推断该陈述对于所有自然数（大于或等于基本情况中指定的数字）都是正确的。形式上，数学归纳法中使用的证明技术是 $^{2}$ 的形式：
$$
(P(1) \wedge \forall k[(P(1) \wedge P(2) \wedge \ldots \wedge P(k)) \rightarrow P(k+1)]) \rightarrow \forall n P( n)
$$
强和弱数学归纳是等价的，任何通过弱数学归纳完成的证明也可以被认为是使用强归纳的证明，并且使用强归纳进行的证明也可以转换为使用弱归纳的证明。

当如何从 $P(k)$ 证明 $P(k+1)$ 相当清楚时，通常使用弱数学归纳法，而在不那么明显的情况下通常使用强数学归纳法。两种形式的数学归纳的有效性都来自于自然数的良序特性，即每个非空集合都有一个最小元素。
良序原则
每个非空自然数集都有一个最小元素。良序原理等价于数学归纳原理。

例 4.3 证明每个大于 1 的自然数都被质数整除。 Proof Base Case 并且可以自行整除。并且可以被自己整除。
感应步骤（强感应）
假设对于每个小于或等于 $k$ 的数，结果都为真。然后我们考虑$k+1$，有两种情况需要考虑。如果$k+1$ 是素数，那么它可以被自身整除。否则，它是复合的，它可以被分解为两个数字的乘积，每个数字都小于或等于 $k$。根据强归纳假设，这些数字中的每一个都可以被一个素数整除，因此 $k+1$ 可以被一个素数整除。

因此，我们已经证明，如果所有小于或等于$k$ 的自然数都可以被一个素数整除，那么$k+1$ 可以被一个素数整除。我们已经证明了基本情况 $\mathrm{P}(2)$ 为真，因此从强大的数学归纳中可以得出，每个大于 1 的自然数都可以被某个素数整除。

数学代写| DISCRETE MATHEMATICS代考请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

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

编码理论（英语：Coding theory）是研究编码的性质以及它们在具体应用中的性能的理论。编码用于数据压缩、加密、纠错，最近也用于网络编码中。不同学科（如信息论、电机工程学、数学、语言学以及计算机科学）都研究编码是为了设计出高效、可靠的数据传输方法。这通常需要去除冗余并校正（或检测）数据传输中的错误。

编码共分四类：^[1]