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A.1. An´ alisis de alternativas de entornos de ex- ex-perimentaci´ on
You are already very familiar with the natural numbers,N D f1; 2; 3; 4; : : : g, which are sometimes called the counting numbers or whole numbers. By adding zero and the negative natural numbers to this set, one obtains the integers,Z D f: : : ; 3; 2; 1; 0; 1; 2; 3; : : : g. The natural numbers are often referred to as the positive integers. Much of a student’s first study of mathematics is concerned with these two sets of numbers. By a very young age most people are already familiar with even and odd integers and some of their properties. This section will construct proofs of some of these properties both because the student will feel very comfortable with the concepts and because it allows for the introduction of some basics about how to write proofs.
Before proceeding with proofs, though, it is necessary that there is agreement on the definitions of even and odd integers. Indeed, there are many possible definitions of even integers:
n2 Z is an even integer if
• the decimal representation of n has a ones digit equal to 0, 2, 4, 6, or 8.
• n is either 0 or the prime factorization of n contains a factor of 2.
• there is an integer k such that n D2k.
• inis a real number, where i Dp
1.
• the number.1/nis positive.
• 9n 1 .mod 10/.
• sin.n2/ D 0.
• n2 2 Z.
Which of these definitions should be used when writing proofs about even and odd integers? Actually, since all the definitions are equivalent, one could adopt any one of these definitions and then prove theorems that show that all the other definitions are equivalent to the chosen definition. This is not an unusual situation in mathematics, especially for a concept as elementary as even integers. But it turns out that one of these definitions is particularly well suited for writing proofs, and that is, n 2 Z is even if there is a k 2 Z such that n D 2k. This makes a useful
definition because it provides a fairly easy way to check whether a given integer is even, and because knowing that a number n is even immediately gives you a number k for which n D2k, and that is a powerful tool for proving facts about even integers.
For this reason, this chosen definition is called the working definition, that is, it is the definition easiest to apply in the wide variety of contexts. It is the definition chosen from which all other properties of even numbers can be derived.
A similar discussion could take place about how to define odd integers. The working definition is that n 2 Z is odd if there is a k 2 Z such that n D 2k C 1.
There is a long list of facts you could prove about even and odd numbers.
Facts About Even and Odd Integers
• Every integer is either even or odd.
• No integer is both even and odd.
• n 2Z is even if and only if n C 1 2 Z is odd.
• The sum of any two even integers is even.
• The sum of any two odd integers is even.
• The sum of an even integer and an odd integer is odd.
• The product of two odd integers is odd.
• The product of two integers is odd only if both of the factors are odd.
Together, the first two of these facts say that each integer is either even or odd but not both. This says that the sets of even and odd integers form a partition of Z, that is, the sets are disjoint and the union of the sets is all of Z. Some authors require that all the sets of a partition be nonempty as in the case with even and odd integers. So why is it that every integer is either even or odd? This depends on the Division Algorithm that states that if m; n 2 Z with n > 0, then there are unique q; r 2 Z with 0 r < n such that m D nq C r. In this case q is called the quotient of the division, and r is called the remainder of the division. Using the Division Algorithm, any integer m can be divided by 2 giving a quotient and remainder where the remainder is either 0 or 1. If the remainder is 0, then m D2q for integer q implying that m is even, and if the remainder is 1, then m D2q C 1 for integer q implying that m is odd.
2.4.2 Proofs About Even and Odd Integers
How can these ideas be used to write a good proof of Every integer is either even or odd? First it is easier to reword the statement as If m 2Z, then either m is even or m is odd. This is a conditional statement, so the natural way to begin a proof is to assume that the hypothesis of the statement is satisfied, that is, that m is an integer. Now apply the Division Algorithm to get the quotient q and remainder r guaranteed by the algorithm. Finally, the value of r shows that m either satisfies the
2.4 Proofs About Even and Odd Integers 29 definition of being an even integer or the definition of being an odd integer. The result would be
PROOF: Ever integer is either even or odd.
• Let m be an integer.
• By the Division Algorithm there are integers q and r with0 r < 2 such that m D2q C r.
• If r D 0, then m D 2q for integer q which means that m satisfies the definition for being even.
• If r D 1, then m D 2q C 1 for integer q which means that m satisfies the definition for being odd.
• Since r must be either 0 or 1, it follows that every integer is either even or odd.
Next consider the how to prove the statement The sum of any two odd integers is even. The statement concerns the sum of any two odd integers, so the proof reader would expect the proof to consider two arbitrarily chosen odd integers. Once two odd integers are chosen, the definition of odd integer should be invoked because, at that point, that is the only information that is known about the two integers. Finally, a little algebra will help to show that the sum of these two odd integers satisfies the definition of even integer. Here is an attempt to write such a proof that makes several common proof writing errors.
PROOF ATTEMPT: The sum of any two odd integers is even.
• The two integers are odd, so each has the form2k C 1.
• The sum of these two integers is.2k C 1/ C .2k C 1/ D 4k C 2.
• k could be even or odd.
• The number 2 is even since it is2 1.
• 4k is even since it is 2 2k.
• The sum of two even numbers is even, so the sum of4k and 2 is an even number.
• Therefore, the sum of two odd integers is always even.
Here are some complaints about the above proof attempt.
• The proof begins talking about two integers, but the proof reader has not yet been introduced to these integers and does not know what two integers are being discussed. The proof is missing a “SET THE CONTEXT” sentence to introduce the idea of starting with any two odd integers.
• The proof uses the variable k without introducing what that variable represents.
The proof requires that k be an integer, but the fact that k is an integer is not stated anywhere. As far as the proof reader knows, k could be any complex number.
Later, the proof claims that2k is an integer which is needed to show 4k is an even integer. Without knowing that k is an integer, it does not follow that2k is also an integer.
• The definition of odd integer allows you to take an odd integer and represent it as 2k C 1, where k is another integer. To apply this definition, then, the proof should
start with an odd integer, say m, and then represent it as2kC1 rather than starting with2k C 1. The subtle point is that one should start with odd integer and use its definition to move on to2k C 1 rather than starting with 2k C 1 which jumps the gun. The reader of the proof could wonder whether2k C 1 could represent a generic odd integer. Well, it can, but this takes some thought which can be avoided by starting with an odd integer m and then using the definition of odd to select the integer k such that m D2k C 1.
• The definition of “odd integer” does refer to “2k C 1,” but it is more precise. It does not say “has the form.” It says that there is an integer k such that the odd number equals2k C 1:
• It is a major error to allow both odd integers to equal2k C 1 for the same number k. The only way this can happen is for the two odd integers themselves to be equal. Thus, this “proof” only applies to a small subset of cases where one adds two identical odd integers together such as3 C 3 or 117 C 117.
• The statement k could be even or odd is certainly correct, but it does not contribute to the proof. It is a statement about items in the proof that is not part of the proof. Occasionally, one will make a definition as part of a long proof, and then give some examples to help the reader understand that definition. But if a statement is not needed either as a critical step in a proof or an important illustration to aid the understanding of the proof, then the statement should be left out of the proof because it distracts from the proof and complicates it.
• The statement The sum of two even numbers is even is correct, but it has not been proved yet, at least in this text, and is equivalent in difficulty to proving the corresponding statement about the sum of odd integers. Thus, it is not appropriate to use the result about sums of even integers to prove one about the sum of odd integers.
Considering these ideas, one can construct a better proof.
PROOF: The sum of any two odd integers is even.
• Let m and n be two odd integers.
• From the definition of odd integer, there is an integer k1 such that m D 2k1C 1 and an integer k2such that n D2k2C 1.
• Then m C n D.2k1C 1/ C .2k2C 1/ D 2.k1C k2C 1/.
• Since k1and k2are integers, so is k1C k2C 1.
• Thus, the sum m C n is equal to twice an integer, so by the definition of even integer, m C n is even.
• Therefore, the sum of any two odd integers is always even.
The form of this proof can be copied almost word for word to get a similar proof of the statement The product of two odd integers is odd.
2.5 Basic Facts About Real Numbers 31
PROOF: The product of any two odd integers is odd.
• Let m and n be two odd integers.
• From the definition of odd integer, there is an integer k1 such that m D 2k1C 1 and an integer k2such that n D2k2C 1.
• Then mn D.2k1C 1/.2k2C 1/ D 4k1k2C 2k1C 2k2C 1 D 2.2k1k2C k1C k2/ C 1.
• Since k1and k2are integers, so is2k1k2C k1C k2.
• Thus, the product mn is equal to one more than twice an integer, so by the definition of odd integer, mn is odd.
• Therefore, the product of any two odd integers is always odd.
2.4.3 Exercises
Write proofs for each of the following statements.
1. The sum of any two even integers is even.
2. The product of an even integer and an odd integer is even.
3. The difference of an even integer and an odd integer is odd.
4. If the product of two integers is odd, then both of the integers must have been odd.
5. The sum of any four consecutive integers is even.