The Choose Method
5.2 USING THE CHOOSE METHOD
During the backward process, if you come across a statement having the quantifier “for all” in the standard form:
B: For all “objects” with a “certain property,” “something happens,”
then one approach to showing that the statement is true is to make the fol-lowing list of all of the objects having the certain property:
Objects with the Certain Property X1
X2
X3
...
Then, for each object on the list, you would need to show that the some-thing happens. When the list consists of only a few objects, this might be a reasonable way to proceed. However, when the list is long or infinite, this approach is not practical. You have already dealt with this type of obstacle in set theory, where the problem is overcome by using set-builder notation to describe the set. Here, the choose method allows you to circumvent the difficulty.
To understand the idea of the choose method consider again the foregoing list of objects, each with the certain property. Suppose you are able to prove that, for object X1, the something happens. Further, imagine that your proof is such that when you replace X1 everywhere with X2, the resulting proof correctly establishes that the something happens for X2. In this case, you do not need to write a separate proof to establish that the something happens for X2. You would simply say, “To see that the something happens for X2, repeat the proof, replacing X1 everywhere with X2.”
Extending this idea to the remaining objects on the list, the goal of the choose method is to construct a “model proof” for establishing that the some-thing happens for a general object X that has the certain property, in such a way that you could, in theory, repeat the proof for each and every object on the list (see Figure 5.1). If you had such a model proof, then you would not need to check the whole (possibly infinite) list of objects because you would know that you could always do so by a simple substitution in the model proof.
In other words, rather than actually proving that the something happens for every object having the certain property, the choose method provides the ca-pability of doing so through the use of a model proof. The way in which this model proof is designed is now described.
To understand how the choose method is used, recall that the model proof must establish that the something happens for a general object having the certain property. As such, suppose that you have one of these objects, say, X, but remember, you do not know precisely which one. All you know is that
5.2 USING THE CHOOSE METHOD 57
Fig. 5.1 A model proof for the choose method.
the particular object X has the certain property. You must somehow use that property to reach the conclusion that, for this object X, the something hap-pens. This is most easily accomplished by working forward from the certain property and backward from the something that happens. In other words, with the choose method,
• You choose an object with the certain property, which then becomes a new statement in the forward process.
• You must show that, for the chosen object, the something happens, which then becomes the new statement in the backward process.
If successful, then you have the capability of repeating the foregoing model proof for any object having the certain property.
An Example of Using the Choose Method
To illustrate how the choose method is used, suppose that, in some proof, you need to show that
B: For all real numbers x with x2− 3x + 2 ≤ 0, 1 ≤ x ≤ 2.
The first step is to identify, in this for-all statement, the object and its type (a real number x), the certain property (x2−3x+2 ≤ 0), and the something that happens (1≤ x ≤ 2). To apply the choose method, you choose a real number that has the certain property. In this case, you might write the following:
A1: Let y be a real number with y2− 3y + 2 ≤ 0.
Then, by working forward from y2−3y +2 ≤ 0, you must reach the conclusion that for y the something happens; that is, you must show that
B1: 1≤ y ≤ 2.
Here, the symbol y is used to distinguish the chosen object from the general object x in B. Notice in A1 and in B1 that the certain property and the something that happens are written for the chosen object, not for the general one. This is done by replacing the general object (x) everywhere in B with the chosen object (y). In many condensed proofs, the same symbol is used for both the general object and the chosen one. In such cases, be careful to interpret the symbol correctly. Consider the following example.
Proposition 6 If S and T are the two sets defined by S = {real numbers x : x2− 3x + 2 ≤ 0}
T = {real numbers x : 1 ≤ x ≤ 2}, then S = T .
Analysis of Proof. When doing a proof, learn to choose a technique consciously, based on the form of the statements under consideration. In this proposition, the hypothesis A and conclusion B do not contain keywords (such as “there is” or “for all”). In the absence of keywords, the forward-backward method is a reasonable technique to use. Doing so in this case gives rise to the key question, “How can I show that two sets (namely, S and T ) are equal?”
Definition 13 provides the answer that you must show that B1: S is a subset of T and T is a subset of S.
So first try to establish that B2: S is a subset of T , and, afterward, that
B3: T is a subset of S.
To show that S is a subset of T (see B2), you obtain the key question,
“How can I show that a set (namely, S) is a subset of another set (namely, T )?” Using Definition 12 leads to the answer that you must show that
B4: For all elements x∈ S, x ∈ T .
This new backward statement, B4, contains the quantifier “for all,” thus indicating that you should proceed by the choose method. To do so, first identify, in B4, the object and its type (an element x ∈ S), the certain property (none) and the something that happens (x∈ T ).
To apply the choose method to B4, you must now choose an object having the certain property and then show that, for this chosen object, the something happens. In this case that means you should choose
A1: An element x∈ S.
5.2 USING THE CHOOSE METHOD 59
Using the fact that x∈ S (that is, that x satisfies the defining property of S) together with the information in A, you must show that, for this chosen object, the something happens in B4; that is,
B5: x∈ T .
Note that you do not want to pick one specific element in S, say 3/2. Also, note the double use of the symbol x for both the general object in B4 and the chosen object in A1.
Working backward from B5, you should ask the key question, “How can I show that an element (namely, x) belongs to a set (namely, T )?” One answer is to show that x satisfies the defining property of T ; that is,
B6: 1≤ x ≤ 2.
Turning now to the forward process, you can make use of the information in A to show that 1≤ x ≤ 2 because you have assumed that A is true. However, additional information is available. Recall that, during the backward process, you used the choose method, at which time you chose x∈ S (see A1). Now is the time to use this fact. Specifically, because x ∈ S, from the defining property of the set S, you know that
A2: x2− 3x + 2 ≤ 0.
Then, by factoring, you obtain A3: (x− 2)(x − 1) ≤ 0.
The only way that the product of x− 2 and x − 1 can be ≤ 0 is for one of the terms to be≤ 0 and the other ≥ 0. In other words,
A4: Either x− 2 ≥ 0 and x − 1 ≤ 0, or else x − 2 ≤ 0 and x− 1 ≥ 0.
The first situation can never happen because, if it did, x ≥ 2 and x ≤ 1, which is impossible. Thus the second condition must happen; that is,
A5: x≤ 2 and x ≥ 1.
But this is precisely the last statement obtained in the backward process (B6), and hence it has been shown successfully that S is a subset of T . Do not forget that you still have to show that T is a subset of S (B3) in order to complete the proof that S = T . This part is done in Section 5.3.
Proof of Proposition 6. To show that S = T , it is shown that S⊆ T and T ⊆ S. To see that S ⊆ T , let x ∈ S. (The use of the word “let” in con-densed proofs frequently indicates that the choose method is being invoked.) Consequently, x2− 3x + 2 ≤ 0 and so (x − 2)(x − 1) ≤ 0. This means that either x− 2 ≥ 0 and x − 1 ≤ 0 or else x − 2 ≤ 0 and x − 1 ≥ 0. The former
cannot happen because, if it did, x≥ 2 and x ≤ 1. Hence it must be that x≤ 2 and x ≥ 1, which means that x ∈ T . The proof that T is a subset of S is given subsequently in Proposition 7.
On a final note, whenever you apply the choose method to choose an object with the certain property, you must first be sure that, indeed, there is at least one such object, for if there are none, how can you choose such an object? To illustrate, suppose you are trying to prove that
B: For all real numbers x≥ 0 with x2+ 3x + 2 = 0, x2≥ 4.
According to the choose method, you should choose a real number x ≥ 0 with the property that x2+ 3x + 2 = 0. However, there are no such numbers (because the only values for x that satisfy the equation are x = −1 and x = −2, and neither of these is ≥ 0). In this case, you cannot apply the choose method; however, there is no need to do so. The reason is that, when there is no object with the certain property, the associated for-all statement is automatically true. To understand why, recall again that, when using the choose method to prove that
S1: For every object with a certain property, something happens, you choose
A: An object X with the certain property, for which you must then show that
B: X satisfies the something that happens.
Observe that this approach is exactly how you would proceed if you were using the forward-backward method to prove that
S2: If X is an object with the certain property, then X satisfies the something that happens.
Specifically, to show that the statement S2 is true with the forward-backward method, you work forward from the hypothesis A given above and backward from the conclusion B given above. In summary, the foregoing statements S1 and S2 are equivalent.
Now you can see why, if there is no object with the certain property, S1 is true. This is because, if there is no object with the certain property, then the hypothesis in S2 is false and so, from Table 1.1 on page 4, the implication in S2 is true. Because S1 is equivalent to S2, S1 is also true. This means that, whenever you need to prove a statement in the form S1, you can prove S2 instead, and vice versa.
5.3 READING A PROOF 61
5.3 READING A PROOF
The process of reading and understanding a proof is now demonstrated.
Proposition 7 If S and T are the two sets defined by S = {real numbers x : x2− 3x + 2 ≤ 0}
T = {real numbers x : 1 ≤ x ≤ 2}, then T ⊆ S.
Proof of Proposition 7. (For reference purposes, each sentence of the proof is written on a separate line.)
S1: To show that T ⊆ S, let t ∈ T . S2: It must be shown that t∈ S.
S3: Because t∈ T , 1 ≤ t ≤ 2, so t − 1 ≥ 0 and t − 2 ≤ 0.
S4: Thus, t∈ S because t2− 3t + 2 = (t − 1)(t − 2) ≤ 0.
The proof is now complete.
Analysis of Proof. An interpretation of statements S1 through S4 follows.
Interpretation of S1: To show that T ⊆ S, let t ∈ T .
The author has worked backward from the conclusion B and asked the key question, “How can I show that a set (namely, T ) is a subset of another set (namely, S)?” Applying Definition 12 means it must be shown that
B1: For all elements x∈ T , x ∈ S.
The author then recognizes the keywords “for all” in the backward statement B1 and uses the choose method to choose an object with the certain property, as indicated by the words “. . . let t∈ T .”
Interpretation of S2: It must be shown that t∈ S.
According to the choose method, it is necessary to show that, for the chosen object, the something happens. This is exactly what the author is saying must be done for the chosen object t, that is, it must be shown that t∈ S.
Interpretation of S3: Because t∈ T , 1 ≤ t ≤ 2, so t − 1 ≥ 0 and t − 2 ≤ 0.
The author is working forward from the fact that t∈ T , using the defining property of T . Presumably this is being done to show that the something happens for the chosen object; that is, that t∈ S.
Interpretation of S4. Thus, t∈ S because t2− 3t + 2 = (t − 1)(t − 2) ≤ 0.
The author is now claiming that t ∈ S by showing that t satisfies the defining property of S. In essence, the author has asked the key question,
“How can I show that an element (namely, t) belongs to a set (namely, S)?”
and has answered the question by using the defining property of the set.
Having shown that t ∈ S, the choose method, and hence the proof, is now complete.
The following points about reading the condensed proof of Proposition 7 are worth noting.
• No mention is made of the techniques being used (the forward-backward and choose method, and the key questions and answers). However, in this case, the word “let” indicates that the choose method is used.
• The techniques used in the proof vary as the form of the statement cur-rently under consideration varies. For example, the author starts with the forward-backward method and then changes to the choose method when a backward statement contains the quantifier “for all.”
• Several steps are condensed into the single sentence S1.
Summary
Use the choose method when the last statement in the backward process contains the quantifier “for all” in the standard form:
For all “objects” with a “certain property,” “something happens.”
To use the choose method, proceed as follows to create a model proof that could, in theory, be repeated for every object with the certain property.
1. Identify the object and its type, the certain property, and the something that happens in the for-all statement.
2. Verify that there is at least one object with the certain property. If there is no such object, then the for-all statement is true and you are done.
3. Choose an object that has the certain property. (Write the fact that the chosen object has the certain property as a new statement in the forward process.)
4. Show that, for this chosen object, the something happens. (Write this objective as the next statement in the backward process.)
Step 4 is accomplished by the forward-backward method. That is, work for-ward from the fact that the chosen object in Step 3 has the certain property and backward from the fact that this chosen object must be shown to satisfy the something that happens. In so doing, you can use the assumption that the hypothesis A, or any other statement in the forward process, is true.
CHAPTER 5: EXERCISES 63
Exercises
Note: Solutions to those exercises marked with a W are located on the web at http://www.wiley.com/college/solow/.
Note: All proofs should contain an analysis of proof and a condensed version.
Definitions for all mathematical terms are provided in the glossary at the end of the book.
W5.1 For each of the following definitions, identify the objects, the certain property, and the something that happens in the for-all statements.
a. The real number x∗is a maximizer of the function f if and only if for every real number x, f(x)≤ f(x∗).
b. Suppose that f and g are functions of one variable. Then g ≥ f on the set S of real numbers if and only if for every element x∈ S, g(x) ≥ f(x).
c. A real number u is an upper bound for a set S of real numbers if and only if for all elements x∈ S, x ≤ u.
5.2 For each of the following definitions, identify the objects, the certain property, and the something that happens in the for-all statements.
a. A function f of one real variable is strictly increasing if and only if for all real numbers x and y with x < y, f(x) < f(y).
b. The set C of real numbers is a convex set if and only if for all elements x, y∈ C, and for every real number t with 0 ≤ t ≤ 1, tx + (1 − t)y ∈ C.
c. The function f of one real variable is a convex function if and only if for all real numbers x and y and for all real numbers t with 0≤ t ≤ 1, it follows that f(tx + (1− t)y) ≤ tf(x) + (1 − t)f(y).
W5.3 Reword the following statements in standard form using the appropriate symbols∀, ∃, ⊃−, as necessary.
a. Some mountain is taller than every other mountain.
b. If t is an angle, then sin(2t) = 2 sin(t) cos(t).
c. The square root of the product of any two nonnegative real numbers p and q is not less than their sum divided by 2.
d. If x and y are real numbers such that x < y, then there is a rational number r such that x < r < y.
5.4 Reword each of the following for-all statements as an equivalent state-ment in the form, “If . . . then . . .”.
a. For every prime number p, p + 7 is composite.
b. For all sets A, B, and C with the property that A⊆ B and B ⊆ C, it follows that A⊆ C.
c. For all integers p and q with q6= 0, p/q is rational.
W5.5 For each of the parts in Exercise 5.1, describe how you would apply the choose method to show that the for-all statement is true. Use a different symbol to distinguish the chosen object from the general object. For instance, for Exercise 5.1(a), to show that x∗ is the maximizer of the function f, you would choose
A1: a real number, say, x0, for which it must then be shown that
B1: f(x0)≤ f(x∗).
5.6 For each of the parts in Exercise 5.2, describe how you would apply the choose method to show that the for-all statement is true. Use a different symbol to distinguish the chosen object from the general object.
5.7 Could you use the choose method to prove each of the following state-ments? Why or why not? Explain. (S is a set of real numbers.)
a. There is an integer n≥ 4 such that n! ≥ 2n. b. For all integers n≥ 4, n! ≥ 2n.
c. If for all elements x∈ S, |x| < 20, then there is an element s ∈ S such that s < 5.
d. If there is an element x∈ S such that x < 5, then for all elements x ∈ S,
|x| < 20.
e. For all real numbers a, b, and c, if 4ac≤ b2, then ax2+ bx + c has real roots.
W5.8 Suppose you are trying to prove that, “If R, S, and T are sets for which R ⊆ S and S ⊆ T , then R ⊆ T .” Write an appropriate key question and answer to create a new statement, B1, in the backward process. Then indicate how the choose method would be applied to B1 by writing a new statement A1 in the forward process that is a result of choosing the appropriate object and a new statement B2 in the backward process indicating what you would have to show about your chosen object. (Do not complete the proof.)
CHAPTER 5: EXERCISES 65
W5.9 Repeat Exercise 5.8 when you are trying to prove that, “If the real number u is an upper bound for a set S of real numbers and the real number v≥ u, then v is an upper bound for S.” [See the definition in Exercise 5.1(c)].
5.10 Repeat Exercise 5.8 when you are trying to prove that, “If the function f(x) = x3, then f is strictly increasing.” [See the definition in Exercise 5.2(a)].
5.11 Repeat Exercise 5.8 when you are trying to prove that, “If f is a convex function and y is a given real number, then {real numbers x : f(x) ≤ y} is a convex set.” [See the definition in Exercise 5.2(b)].
5.12 Repeat Exercise 5.8 when you are trying to prove that, “If f and g are convex functions, then the function f + g is a convex function.” [See the definition in Exercise 5.2(c)].
W5.13 Suppose you are trying to prove that, “If S and T are the sets defined by S = {(x, y) : x2+ y2 ≤ 16} and T = {(x, y) : 3x2+ 2y2 ≤ 125}, then for every element (x, y)∈ S, (x, y) ∈ T .” Which of the following constitutes a correct application of the choose method? For those that are incorrect, explain what is wrong.
a. Choose
A1 : real numbers x0 and y0. It must be shown that
B1 : (x0, y0)∈ T . b. Choose
A1 : real numbers x0 and y0 with (x0, y0)∈ S.
It must be shown that B1 : (x0, y0)∈ T . c. Choose
A1 : real numbers x0 and y0 with (x0, y0)∈ T . It must be shown that
B1 : (x0, y0)∈ S.
d. Choose
A1 : real numbers x and y, say 1 and 2, with x2+ y2= 5≤ 16 and therefore (x, y) ∈ S.
It must be shown that
B1 : 3x2+ 2y2≤ 125 and therefore that (x, y) ∈ T . e. Choose
A1 : real numbers x and y with (x, y)∈ S.
It must be shown that B1 : (x, y)∈ T .
5.14 Suppose you are trying to prove that, “If p(x) = a0+a1x1+· · ·+anxn is a polynomial of degree n > 1 such that ai > 0 for all integers i = 0, . . . , n, then for all real numbers x and y with 0 < x < y, p(x) < p(y).” Which of the following constitutes a correct application of the choose method? For those that are incorrect, explain what is wrong.
5.14 Suppose you are trying to prove that, “If p(x) = a0+a1x1+· · ·+anxn is a polynomial of degree n > 1 such that ai > 0 for all integers i = 0, . . . , n, then for all real numbers x and y with 0 < x < y, p(x) < p(y).” Which of the following constitutes a correct application of the choose method? For those that are incorrect, explain what is wrong.