Write Your Own Proofs. Amy Babich

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       ∉ .

       ∈ .

      Remarks. We do not describe the sets and more precisely in this chapter because so far we lack the notation to do so. In some books, 0 is regarded as an element of . In this book, we adhere to the tradition that 0 ∉ . We will say more about and in Chapter 3.

      Quantifiers. Quantifiers are important mathematical tools. Using quantifiers, we can make our mathematical language precise. There are two principal quantifiers in mathematics: the universal and the existential.

      Universal quantifier. The universal quantifier has the symbolic form ∀. To express the universal quantifier in English, we write “for all,” “for every,” or “for each.”

      Examples (1.5) The following statements are equivalent.

       1. (∀x ∈ )(x + 1 ∈ )

       2. For every x in , x + 1 is in .

       3. For every natural number x, the number x + 1 is also a natural number.

       4. Given any natural number x, x + 1 is a natural number.

      Remark. Of course, there are other ways to state the sentence: “For all x ∈ , x + 1 ∈ .” For example, we can say, “If you add 1 to a natural number, you get a natural number.” There is nothing wrong with this sentence, but it is not standard “mathematical English.” That is, it is not the language of sets and quantifiers. Mathematical diction would sound peculiar in a non-mathematical context. But such language is very useful for expressing mathematical statements.

      Existential quantifier. The existential quantifier has the symbolic form ∃. To express the existential quantifier in words, we say “there exists” or “there is” or “for some” or “there is at least one.”

      Examples (1.6) The following statements axe equivalent.

       1. (∃x ∈ )(x > 5)

       2. There exists a natural number x such that x > 5.

       3. There is at least one natural number greater than 5.

       4. For some natural number x, the number x is greater than 5.

      The phrase “there exists [some object]” is often followed by “such that.” The phrase “such that” is used in mathematics instead of phrases involving the relative pronouns “which,” “that,” or “whose.”

      Order of quantifiers. The order of quantifiers in a sentence is important. The following examples illustrate this point.

      Examples (1.7) Consider these two statements:

       1. (∀x ∈ )(∃y ∈ )(y > x)

      For each x ∈ , there exists y ∈ such that y > x.

       2. (∃y ∈ )(∀x ∈ )(y > x)

      There exists y ∈ such that for each x ∈ , y > x.

      Statement 1 says that given any positive integer, there is a larger positive integer. Statement 1 is true.

      Since “∀x ∈ ” comes before “∃y ∈ ,” y depends on x. For different values of x there are different values of y.

      Let x = 5. There exists y ∈ such that y > x. For example, 6 > 5.

      Let x = 6. There exists y ∈ such that y > 6. For example, 10 > 6.

      Statement 2 says that there is a positive integer which is larger than every positive integer, including itself.

      Statement 2 is false.

      Since “∃y ∈ ” comes before “∀x ∈ ,” the value of y does not depend on x. The statement says that there is one number y that works for all natural numbers x.

      Exercises (1.5) Write out each statement using words rather than symbols. Then classify the statements either true or false. Explain your answers.

       1. (∃x ∈ )(∀y ∈ )(x + y = 0)

       2. (∀y ∈ )(∃x ∈ )(x