Week 5: Quantifiers Quantity has a quality all its own. attributed to Carl von Clausewitz Number Sets Many people would say that mathematics is “the science of numbers.” 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. Almost all of what we have done so far has involved numbers – mostly whole numbers. The most basic kind of number 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 set of numbers1 is usually abbreviated by the symbol ℕ. The next most basic kind of number are the integers, 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 “ratios” (i.e. quotients) 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 positive real numbers, 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 point 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 integer is either even or odd. 1 Some textbooks do not consider 0 to be a natural number, 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 quantifier, 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 order 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 real number 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.