The Choose Method
5.1 WORKING WITH THE QUANTIFIER “FOR ALL”
Quantifiers II:
The Choose Method
This chapter develops a technique for dealing with statements in the backward process that contain the quantifier “for all.” Such statements arise quite naturally in many mathematical areas, one of which is set theory, as you will now see.
5.1 WORKING WITH THE QUANTIFIER “FOR ALL”
A set is a collection of items. For example, you can think of the numbers 1, 4, and 7 as a collection of items, and hence they form a set. Each of the individual items is called a member or element of the set and each member of the set is said to be in or belong to the set. The set is often denoted by enclosing the list of its members, separated by commas, in braces. Thus, the set consisting of the numbers 1, 4, and 7 is written as follows:
{1, 4, 7}.
To indicate that the number 4 belongs to this set, mathematicians write:
4∈ {1, 4, 7},
where the symbol ∈ stands for the words “is a member of.” Similarly, to indicate that 2 is not a member of{1, 4, 7}, one would write:
26∈ {1, 4, 7}.
While it is desirable to make a list of all the elements in a set, sometimes it is impractical to do so because the list is too long. For example, imagine 53
having to write down every integer between 1 and 100,000. When a set has an infinite number of elements (such as the set of real numbers that are greater than or equal to 0) it is impossible to make a complete list, even if you want to. Fortunately there is a way to describe such “large” sets through the use of what is known as set-builder notation, which involves using a verbal and mathematical description for the members of the set. With set-builder notation, the set of all real numbers that are greater than or equal to 0 is written as follows:
S ={real numbers x : x ≥ 0},
where the “:” stands for the words “such that.” Everything following the
“:” is referred to as the defining property of the set. The question that one always has to be able to answer is, “How do I know if a particular item, say y, belongs to the set?” To answer such a question, you need only check if the item y satisfies the defining property. If so, then y is an element of the set; otherwise, y is not in the set. For the foregoing set S, to see if the real number 3 belongs to S, simply replace x everywhere by 3 and see if the defining property is true. In this case, 3 does belong to S because 3≥ 0.
Sometimes part of the defining property appears to the left of the “:” as well as to the right, so, when trying to determine if a particular item belongs to such a set, be sure to verify this portion of the defining property, too. For example, if T = {real numbers x ≥ 0 : x2− x + 2 ≥ 0}, then −1 does not belong to T even though−1 satisfies the defining property to the right of the
“:”. The reason is that−1 does not satisfy the defining property to the left of the “:” because −1 is not ≥ 0.
From the point of view of doing proofs, the defining property plays the same role as a definition—the defining property is used to answer the key question, “How can I show that an item belongs to a particular set?” One answer is to check that the item satisfies the defining property.
While discussing sets, observe that it can happen that no item satisfies the defining property. Consider, for example,
{real numbers x ≥ 0 : x2+ 3x + 2 = 0}.
The only real numbers for which x2+ 3x + 2 = 0 are−1 and −2. Neither of these values satisfies the defining property to the left of the “:”. Such a set is said to be empty, meaning that the set has no members. The special symbol
∅ is used to denote the empty set.
To motivate the use of the quantifier “for all,” observe that it is usually possible to write a set in more than one way—for example, the two sets
S = {real numbers x : x2− 3x + 2 ≥ 0}, T = {real numbers x : 1 ≤ x ≤ 2},
where 1≤ x ≤ 2 means that 1 ≤ x and x ≤ 2. Surely for two sets S and T to be the same, each element of S should appear in T and vice versa. Using the quantifier “for all,” a definition can now be made.
5.1 WORKING WITH THE QUANTIFIER “FOR ALL” 55
Definition 12 A set S is a subset of a set T (written S⊆ T or sometimes S⊂T ) if and only if for each element x ∈ S, x ∈ T .
Definition 13 Two sets S and T are equal (written S = T ) if and only if S is a subset of T and T is a subset of S.
Like any definition, these are used to answer a key question. Definition 12 answers the question, “How can I show that a set (say, S) is a subset of another set (say, T )?” by requiring you to show that for each element x∈ S, x∈ T . How you do so is explained in Section 5.2. Definition 13 answers the key question, “How can I show that two sets (say, S and T ) are equal?” by requiring you to show that S is a subset of T and T is a subset of S.
In addition to set theory, there are many other instances where the quan-tifier “for all” is used, but, from the foregoing example, you can see that such statements appear to have the same consistent structure. When the quantifier
“for all,” “for each,” “for every,” or “for any” appears, the statement will have the following standard form (which is similar to the one in Chapter 4):
For every “object” with a “certain property,” “something happens.”
The words in quotation marks depend on the particular statement under consideration, and you must learn to read, to write, and to identify these three components—keeping in mind to identify the type of object. Consider these examples.
1. For every angle t, sin2(t) + cos2(t) = 1.
Object: angle t.
Certain property: none (there might not be a certain property).
Something happens: sin2(t) + cos2(t) = 1.
Mathematicians often use the symbol∀ to abbreviate the words “for all” (“for each,” and so on). The use of symbols is illustrated in the next example.
2. ∀ real numbers y > 0, ∃ a real number x ⊃− 2x= y.
Object: real numbers y.
Certain property: y > 0.
Something happens: ∃ a real number x ⊃− 2x= y.
Observe that a comma always precedes the something that happens.
Sometimes the quantifier is hidden; for example, the statement “the cosine of an angle strictly between 0 and π/4 is larger than the sine of the angle” could be phrased equally well as “for every angle t with 0 < t < π/4, cos(t) > sin(t).”
Also, some authors write the quantifier after the something that happens; for example: “2n > n2, for all integers n ≥ 5.” Practice is needed to become fluent at reading and writing these statements, regardless of how they are presented, keeping in mind that the constituent parts could be implied from the context, out of order, or not present.