Determine which of the statement are true and which are false. Prove each true statement d...

Determine which of the statement are true and which are false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. In case the statement is false, determine whether a small change would make it true. If so, make the change and prove the new statement. Follow the directions for writing proofs.

Directions for Writing Proofs of Universal Statements

Think of a proof as a way to communicate a convincing argument for the truth of a mathematical statement. When you write a proof, imagine that you will be sending it to a capable classmate who has had to miss the last week or two of your course. Try to be clear and complete. Keep in mind that your classmate will see only what you actually write down, not any unexpressed thoughts behind it. Ideally, your proof will lead your classmate to understand why the given statement is true.

Over the years, the following rules of style have become fairly standard for writing the final versions of proofs:

1. Copy the statement of the theorem to be proved on your paper.

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2. Clearly mark the beginning of your proof with the word Proof.

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3. Make your proof self-contained.

This means that you should explain the meaning of each variable used in your proof in the body of the proof. Thus you will begin proofs by introducing the initial variables and stating what kind of objects they are. The first sentence of your proof would be something like “Suppose m and n are any even integers” or “Let x be a real number such that x is greater than 2.” This is similar to declaring variables and their data types at the beginning of a computer program.

At a later point in your proof, you may introduce a new variable to represent a quantity that is known at that point to exist. For example, if you have assumed that a particular integer n is even, then you know that n equals 2 times some integer, and you can give this integer a name so that you can work with it concretely later in the proof. Thus if you decide to call the integer, say, s, you would write, “Since n is even, n = 2s for some integer s,” or “since n is even, there exists an integer s such that n = 2s.”

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4. Write your proof in complete, gramatically correct sentences.

This does not mean that you should avoid using symbols and shorthand abbreviations, just that you should incorporate them into sentences. For example, the proof of Theorem contains the sentence

Then m + n = 2r + 2s

= 2(r + s).

To read such text as a sentence, read the first equals sign as “equals” and each subsequent equals sign as “which equals.”

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5. Keep your reader informed about the status of each statement in your proof.

Your reader should never be in doubt about whether something in your proof has been assumed or established or is still to be deduced. If something is assumed, preface it with a word like Suppose or Assume. If it is still to be shown, preface it with words like,We must show that or In other words, we must show that. This is especially important if you introduce a variable in rephrasing what you need to show.

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4. Write your proof in complete, gramatically correct sentences.

This does not mean that you should avoid using symbols and shorthand abbreviations, just that you should incorporate them into sentences. For example, the proof of Theorem contains the sentence

Then m + n = 2r + 2s

 = 2(r + s).

To read such text as a sentence, read the first equals sign as “equals” and each subsequent

equals sign as “which equals.”

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5. Keep your reader informed about the status of each statement in your proof.

Your reader should never be in doubt about whether something in your proof has been assumed or established or is still to be deduced. If something is assumed, preface it with a word like Suppose or Assume. If it is still to be shown, preface it with words like,We must show that or In other words, we must show that. This is especially important if you introduce a variable in rephrasing what you need to show.

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6. Give a reason for each assertion in your proof.

Each assertion in a proof should come directly from the hypothesis of the theorem, or follow from the definition of one of the terms in the theorem, or be a result obtained earlier in the proof, or be a mathematical result that has previously been established or is agreed to be assumed. Indicate the reason for each step of your proof using phrases such as by hypothesis, by definition of …, and by theorem …

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7. Include the “little words and phrases” that make the logic of your arguments clear.

When writing a mathematical argument, especially a proof, indicate how each sentence is related to the previous one. Does it follow from the previous sentence or from a combination of the previous sentence and earlier ones? If so, start the sentence by stating the reason why it follows or by writing Then, or Thus, or So, or Hence, or Therefore, or Consequently, or It follows that, and include the reason at the end of the sentence. For instance, in the proof of Theorem, once you know that m is even, you can write: “By definition of even, m = 2r for some integer r ,” or you can write, “Then m = 2r for some integer r by definition of even.”

If a sentence expresses a new thought or fact that does not follow as an immediate consequence of the preceding statement but is needed for a later part of a proof, introduce it by writing Observe that, or Note that, or But, or Now.

Sometimes in a proof it is desirable to define a new variable in terms of previous variables. In such a case, introduce the new variable with the word Let. For instance, in the proof of Theorem, once it is known that m + n = 2(r + s), where r and s are integers, a new variable t is introduced to represent r + s. The proof goes on to say, “Let t = r + s. Then t is an integer because it is a sum of two integers.”

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8. Display equations and inequalities.

The convention is to display equations and inequalities on separate lines to increase readability, both for other people and for ourselves so that we can more easily check our work for accuracy. We follow the convention in the text of this book, but in order to save space, we violate it in a few of the exercises and in many of the solutions contained in Appendix B. So you may need to copy out some parts of solutions on scratch paper to understand them fully. Please follow the convention in your own work. Leave plenty of empty space, and don’t be stingy with paper!

Theorem

The sum of any two even integers is even.

Proof:

Suppose m and n are [particular but arbitrarily chosen] even integers. [We must show that m + n is even.] By definition of even, m = 2r and n = 2s for some integers r and s. Then

m + n = 2r + 2s by substitution

 = 2(r + s) by factoring out a 2.

Let t = r + s. Note that t is an integer because it is a sum of integers. Hence

m + n = 2t where t is an integer.

It follows by definition of even that m + n is even. [This is what we needed to show.]

Exercise

For all real numbers a and b, if a < b then a <  < b.

(You may use the properties of inequalities in T17-T27.)

Properties of inequalities

T17. Trichotomy Law For arbitrary real numbers a and b, exactly one of the three relations a or a = b holds.

T18. Transitive Law If a and b, then a.

T19. If a, then a + c + c.

T20. If a and c > 0, then ac.

T21. If a _= 0, then a2 > 0.

T22. 1 > 0.

T23. If a and c<0, then ac > bc.

T24. If a, then −a > −b. In particular, if a<0, then −a > 0.

T25. If ab > 0, then both a and b are positive or both are negative.

T26. If a and b, then a + b + d.

T27. If 0 < a and 0 < b, then 0 < ab.