## 数学代写|图论作业代写Graph Theory代考|MATH361

2023年3月27日

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## 数学代写|图论作业代写Graph Theory代考|Proof Techniques

Graph Theory as a mathematical discipline straddles the distinction between applied and theoretical mathematics. As we’ve already seen in this chapter, graphs can be used to model various scenarios, especially some complicated modern applications. Alternatively, isomorphisms comprise a very theoretical aspect of graph theory. Within each of these areas, however, we use proofs to deepen and demonstrate our understanding of graphs and their properties.
While this book relies on a basic understanding of logic, proof structure, and proof techniques, it is by no means expected that the reader is a proficient writer of proofs. This section is meant to review the basics of mathematical proof, and introduce some early graph results that can be proven with little intuition about graphs and their structure. For a more complete introduction to logic and proofs, see Discrete Mathematics by Susanna Epp [31].

Most mathematical statements have an underlying conditional form; that is, they can be written as “If …, then ….” For example, we may say “The sum of two odd integers is even” but we are, in fact, making the conditional statement “If $x$ and $y$ are odd integers, then $x+y$ is even.” Writing a statement in the standard if-then form allows the logical structure to stand out and provides guidance into the format of the argument.

In logical symbols, conditional statements are given as $p \rightarrow q$. A direct proof begins by assuming the premise of the conditional $(p)$ and uses logic, definitions, and previously proven theorems to show the conclusion $(q)$ is true. The example below uses the definition of odd, even, and the assumption that the sum of two integers is still an integer.

## 数学代写|图论作业代写Graph Theory代考|Indirect Proofs

Direct proofs can be considered the preferable method of proof as their structure models the statement they are proving. However, some statements are either impossible or much more difficult to prove in this way and a different technique is needed. Classic examples of this include proving there are infinitely many primes or that $\sqrt{2}$ is irrational. While these are great examples, better examples exist in graph theory for the usefulness of indirect proofs.
There are two main types of indirect proofs: contradiction and contraposition. For a Proof by Contradiction, we assume the negation of the statement is true. Through logic, definitions, and previous results, we show a contradiction must be occurring, thus proving the original statement must be true. An example from elementary number theory is shown below.
Proposition 1.23 For any integer $n$, if $n^2$ is odd then $n$ is odd.
Proof: Suppose for a contradiction that $n^2$ is odd but $n$ is even. Then $n=2 k$ for some integer $k$ and $n^2=(2 k)^2=4 k^2=2 j$ where $j$ is the integer $2 k^2$. Thus $n^2$ is both even and odd, a contradiction. Therefore if $n^2$ is odd then $n$ is also odd.

For a Proof by Contraposition, we use a direct proof on the contrapositive $(\sim q \rightarrow \sim p)$ of the original conditional statement $(p \rightarrow q)$. Since the contrapositive is logically equivalent to the original statement, this shows the intended result to be true. The statement above can also be proven using the contrapositive, as shown below.

Proof: Suppose $n$ is not odd. Then $n$ is even and $n=2 k$ for some integer $k$. Then $n^2=(2 k)^2=4 k^2=2 j$ where $j$ is the integer $2 k^2$, and so $n^2$ is even. Thus if $n^2$ is odd, it must be that $n$ is also odd.

# 图论代考

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