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Week 5: Quantifiers

Quantity has a quality all its own.

attributed to Carl von Clausewitz

Number Sets Many people would say that is “the science of .” This is a common misconception among those who are unfamiliar with the most modern parts of math – that is, ideas that have been developed since the nineteenth century or so. But it is true that most of mathematics does have some connection to numbers. all of what we have done so far has involved numbers – mostly whole numbers. The most basic kind of is referred to as a “natural” number. The natural numbers are the whole numbers starting from 0, 1, and so on. As a whole, this of numbers1 is usually abbreviated by the symbol ℕ. The next most basic kind of number are the , which are all of the whole numbers, including the natural numbers and also the negative whole numbers. As a set, the integers are usually represented by the symbol ℤ. Another type of number are the rational numbers (often simply called the rationals), which are all of the possible “” (i.e. ) of integers. This set of numbers is usually represented2 by the symbol ℚ. As we have seen firsthand, there are numbers with real, physical significance that are not rational. The set of numbers that includes the rationals and the irrationals is known as the real numbers, or simply the reals, and is usually represented by the symbol ℝ. Lastly, it is often useful to refer to the set of all , represented by the symbol ℝ+. Likewise, the set of all positive integers is often represented by the symbol ℤ+. Much of what we did in Weeks 1 through 4 dealt only with the set of positive integers.

Quantifier Symbols Most mathematical statements are “quantified” in some way, and it will become increasingly important to pay attention to this aspect of mathematics. The majority of the statements we have seen up to this have been “universal” in some sense. As a simple example, we once proved that every positive whole number is either even or odd; indeed, going beyond this, it is possible to prove that every is either even or odd.

1 Some textbooks do not consider 0 to be a , but we will take the very slightly more popular approach of including 0. 2 It should be clear what the letters ℕ and ℚ stand for. The letter ℤ stands for “Zahlen,” which is the German word for “numbers”. Week 5 Page 2

A statement that expresses a truth about every member of a set – for example, every integer – is sometimes said to be universal. “Every integer is even or odd” is one such statement. A special symbol is sometimes used to represent that a statement is universal. For example, we might write (∀푥 ∈ ℤ)(푥 is even ∨ 푥 is odd) for the statement mentioned before. The symbol “∀” is most commonly pronounced as “for all,” and the symbol “∈” is usually pronounced as “in.” So another way to say the statement above is “For all 푥 in the set of integers, 푥 is even or 푥 is odd.” It is important to note that the letter 푥 used in this statement is completely irrelevant; for example, the statement (∀푁 ∈ ℤ)(푁 is even ∨ 푁 is odd) is utterly identical in meaning to the statement above. The “∀” symbol is known as a , of which there are two main types. To illustrate the other, slightly less common type, consider the statement “40 is a multiple of 8,” which really means “There is an integer 푚 such that 40 = 8푚.” This is an example of an existential statement, because it claims that a certain object exists. A special symbol is sometimes used to signify existence. For example, we could write (∃푚 ∈ ℤ)(40 = 8푚) for the statement above. The symbol “∃” is most commonly pronounced as “there exists.” So another way to say the statement above is “There exists an 푚 in the set of integers such that 40 = 8푚.” Again, the letter 푚 used in this statement is completely unimportant, and could be replaced with any other reasonable letter as a placeholder. It is extremely common for mathematical statements to involve two or more quantifiers, usually from each of the two main types. For example, the statement “Every integer is even or odd” could be expressed in fully symbolic form as follows: (∀푥 ∈ ℤ)(∃푚 ∈ ℤ)(푥 = 2푚 ∨ 푥 = 2푚 − 1). That is, “For every integer 푥, there is an integer 푚 such that 푥 is either 2푚 or 2푚 − 1.” Perhaps surprisingly, the in which two quantifiers appear in a statement can significantly change the meaning (and truth) of the statement. For example, consider the following statement: (∀푥 ∈ ℕ)(∃푦 ∈ ℕ)(푦 ≥ 푥). For each natural number 푥, there is a natural number 푦 such that 푦 ≥ 푥. This is a true statement. On the other hand, consider the following: (∃푦 ∈ ℕ)(∀푥 ∈ ℕ)(푦 ≥ 푥). This claims that there is a natural number 푦 such that, for each natural number 푥, 푦 ≥ 푥. This is certainly not true! Week 5 Page 3

There is a third, much less common type of quantifier used to indicate a unique existential statement. To illustrate, consider the fact that there is exactly one that, when added to any real number, does not change it. This can be written as (∃! 푦 ∈ ℝ)(∀푥 ∈ ℝ)(푥 + 푦 = 푥) The symbol “∃!” is most commonly pronounced as “there exists a unique…” So we could say the statement above as “There exists a unique real number 푦 such that, for every real number 푥, 푥 + 푦 is 푥.” This kind of statement indicates that a certain object exists, and that there is no other object with the same property. For minimal ambiguity, the usual existence quantifier, “∃,” can be pronounced as “there exists at least one,” while the quantifier “∃!” can be pronounced as “there exists exactly one.” Many true existential statements would also be true if written as unique existential statements. For example, the following statements are both true: (∃! 푚 ∈ ℤ)(40 = 8푚). (∀푥 ∈ ℤ)(∃! 푚 ∈ ℤ)(푥 = 2푚 ∨ 푥 = 2푚 − 1). However, the following statement is not true. (∀푥 ∈ ℕ)(∃! 푦 ∈ ℕ)(푦 ≥ 푥) We do not normally indicate unique existence unless we have some good reason to do so. This is partly because unique existence is more difficult to prove, and in most situations, there is no practical reason to distinguish between existence and unique existence.

Negations Consider the statement “8 is not a multiple of 3.” This means that there is no integer 푚 such that 8 = 3푚. So we can write this statement as ¬[(∃푚 ∈ ℤ)(8 = 3푚)]. Another way to express the same statement is this: for any integer 푚, 8 cannot be 3푚. That is, (∀푚 ∈ ℤ)(8 ≠ 3푚). On a similar note, consider the statement “Not all natural numbers are positive.” This could be written as ¬[(∀푥 ∈ ℕ)(푥 > 0)]. Another way to say this is “There exists a natural number 푥 such that 푥 is not positive.” That is, (∃푥 ∈ ℕ)(푥 ≯ 0). Week 5 Page 4

(Of course, the usual way to write “푥 ≯ 0” is “푥 ≤ 0.”) These two examples illustrate a general rule: the negation of a universal statement is an existential statement, and the negation of an existential statement is a universal statement. Consider the statement “No integer can be both even and odd.” This could be written as follows: ¬[(∃푛 ∈ ℤ)(∃푎 ∈ ℤ)(∃푏 ∈ ℤ)(푛 = 2푎 ∧ 푛 = 2푏 − 1)]. In English, “It is not true that there are integers 푛, 푎, and 푏 such that 푛 = 2푎 and 푛 = 2푏 − 1.” Alternatively, the same statement can be expressed as (∀푛 ∈ ℤ)(∀푎 ∈ ℤ)(∀푏 ∈ ℤ)(푛 ≠ 2푎 ∨ 푛 ≠ 2푏 − 1). “For any integers 푛, 푎, and 푏, either 푛 is not 2푎, or 푛 is not 2푏 − 1.”

Exercise 1: For each of the following quantified statements, write the negation in such a way that the “¬” symbol is not needed. (a) (∀푎 ∈ ℝ)(∀푏 ∈ ℤ)(푎2 + 푏 ∈ ℤ) (b) (∃푦 ∈ ℝ)(∀푥 ∈ ℝ)(푥 ⋅ 푦 = 푥) (c) (∀푥 ∈ ℤ)(∃푦 ∈ ℝ)(푥 = 2푦) (d) (∀푥 ∈ ℤ)(∃푦 ∈ ℝ)(푥/푦 = 2) (e) (∀푥 ∈ ℝ)(∀푑 ∈ ℝ+)(∃푦 ∈ ℚ)(|푥 − 푦| < 푑) Exercise 2: For each of the statements in the previous exercise, determine whether the original statement, or the negation you wrote, is true. Prove each of your answers, or at least explain them as well as you can. Exercise 3: Consider the statement “The graphs of 푦 = 푥2 + 푥 + 2 and 푦 = 푥 − 2 do not intersect.” (a) Write this statement in fully symbolic form (i.e. without any words), using quantifiers, and in such a way that the “¬” symbol appears at the beginning of the statement. (b) Rewrite the statement in fully symbolic form (again, using quantifiers and no words), without using the “¬” symbol. Exercise 4: Write the negation of each of the following statements as an English sentence. Do not use any quantifier symbols, logical symbols, or the word “not.” (a) For any odd integer 푏, there is no real number 푥 such that 푥2 + 푏푥 + 15 = 0. (b) For every pair of real numbers 푥 and 푦, there is an integer 푛 such that 푥 < 푛 < 푦. (c) For every pair of integers 푥 and 푦, there is a real number 푛 such that 푥 < 푛 < 푦. (d) For every pair of integers that sum to 5, one of the numbers must be larger than 2. Week 5 Page 5

Exercise 5: For each of the statements in the previous exercise, determine whether the original statement, or the negation you wrote, is true. Prove each of your answers. Exercise 6: Consider the statements in Exercise 1, parts (b), (c), and (e). Imagine we replaced the existential quantifier in each statement with a unique existential quantifier. For each of the resulting three statements, determine whether it is true, and briefly explain your answer. Exercise 7: Determine how to write the negation of the statement (∃! 푚 ∈ ℤ)(40 = 8푚), in such a way that the “¬” symbol is not needed.

Quantifiers and A central part of proving anything in mathematics is agreeing on the definitions of mathematical words. For example, before proving anything about even and odd numbers, we had to settle upon definitions for the words “even” and “odd.” When defining a mathematical concept, we have somewhat different standards than we might have when defining an everyday concept. If you were to ask a nonmathematician to define an even number, he or she might reply, “Even numbers are ones like 2, 4, 6, 8, and so on.” This has the advantage of being straightforward and understandable, which are usually important qualities for a . However, this statement is not suitable as a mathematical definition. The most obvious problem is that it does not make clear whether nonpositive numbers are allowed to be even. While everyday definitions are intuitive but often not exact, mathematical definitions are exact but not always intuitive. By “exact,” I mean that any given object either satisfies the definition or does not. A proper mathematical definition must leave no opportunity for personal judgement. It must be noted here that there is no single correct definition of any mathematical concept. Just as different board games might come with different pieces, different of mathematics (or even different individual ) may use different words, or different definitions of the same word. This is part of why we will take care to define the mathematical words we use very carefully, and use the definitions consistently. It is most proper, logically speaking, to state most definitions as a biconditional “if and only if” statement3. For example, the meaning of the word “even” that we have been working with has been this: To say that a number is “even” means that the number is a multiple of 2. This wording, used some ago, is a little imprecise; it could be interpreted to mean “If a number is ‘even,’ then it is a multiple of 2.” What we actually want to say is “A number is even if and only if it is a multiple of 2.”

3 Remember from Week 3 that the phrase “if and only if” is often abbreviated as “iff.” Week 5 Page 6

Like other types of mathematical statements, virtually all definitions involve quantifiers. We now state the definitions of various words that we have used before. Note that the following definitions are to be used for the of the course. Definition: 푥 is even iff (∃푚 ∈ ℤ)(푥 = 2푚). Definition: 푥 is odd iff (∃푚 ∈ ℤ)(푥 = 2푚 + 1). This new and final definition of “odd” has been changed here, compared to Week 1. This new definition is more commonly used, and will be more convenient in the future. Definition: 푥 is a iff (∃푎 ∈ ℕ)(∃푏 ∈ ℕ)(푎 ≠ 푥 ∧ 푏 ≠ 푥 ∧ 푎푏 = 푥). You will prove below that a composite number must be a natural number greater than 1. Definition: 푥 is a iff 푥 is a natural number greater than 1, and 푥 is not composite. Definition: 푥 is a iff (∃푎 ∈ ℤ)(∃푏 ∈ ℤ)(푏 ≠ 0 ∧ 푥 = 푎/푏). Recall that the set of all rational numbers is given the symbol ℚ. Definition: 푥 is an iff 푥 is in ℝ and 푥 is not in ℚ.

Exercise 8: State the definition of the word “multiple,” with quantifier(s), in symbolic form. Exercise 9: For the sake of simplicity, in Week 1, most of the mathematical propositions we dealt with involved positive whole numbers – i.e. members of ℤ+. The definition of “odd” given in Week 1 was somewhat different from the definition given here. State that definition in symbolic form, and explain why it was necessary to use that definition when working exclusively with positive integers like 1, 2, 3, etc. Exercise 10: State the definition of the phrase “ number” (as it was used in Week 1), using quantifier(s), in symbolic form. Exercise 11: Prove that every composite number must be a natural number, but cannot be 0 or 1. Exercise 12: Use quantifiers, in symbolic form, to complete the following definition. A rational number 푎/푏 is a reduced iff ……………………………………………..