Write Your Own Proofs. Amy Babich

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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 + y = 0)

       3. (∀x ∈ )(∃y ∈ )(xy = x)

       4. (∃y ∈ )(∀x ∈ )(xy = x)

       5. (∀x ∈ )(∃y ∈ )(x = y – 7)

       6. (∃y ∈ )(∀x ∈ )(x = y – 7)

       7. (∀y ∈ )(∃x ∈ )(x = y –7)

       8. (∀x ∈ )(∀y ∈ )(x = y – 7)

      Remarks. When translating an English sentence into logical symbols, always place a quantifier before the statement it governs. English sentences have various ways of expressing quantifiers. For example, consider the sentence: “Any rational number can be expressed as a fraction whose numerator is an integer and whose denominator is a natural number.” This sentence can be written symbolically as follows:

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