Write Your Own Proofs. Amy Babich

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       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: The same sentence can be paraphrased in mathematical English as follows: For each x ∈ , there exist a ∈ and b ∈ such that

      The statement means the same thing as the statement The names of the variables do not matter. Only their roles in the sentence matter.

      Exercises (1.6) Write each of the following using quantifiers and symbols. In Exercises 9 and 10, the symbol ε is pronounced “epsilon,” with the accent on the first syllable and all vowels short. The symbol δ is pronounced “delta.”

       1. For all integers x and y, the numbers xy and yx are equal.

       2. Given any real number x, there exists a natural number n such that x < n.

       3. Given any real number x, there exists a natural number y such that x + y = 0.

       4. Given any nonnegative real number x, there exists a natural number y such that y² = x.

       5. Given any nonzero real number x, there exists a natural number y such that xy = 1.

       6. There exists a smallest natural number.

       7. There is no largest integer.

       8. Given any two distinct real numbers, some rational number lies strictly between them.

       9. Given any positive real number ε, there exists a natural number k such that whenever n is a natural number greater than k.

      10. For each real number ε, if ε > 0 then there exists a positive real number δ such that for each real number x, if |x – 2| < δ then |x² – 4| < ε.

      Negating quantified statements. Let A be a set, and for each x ∈ A let p(x) be a statement.

      Consider statement (a) below.

      (a) ¬(∀x ∈ A)(p(x))

      It is false that for all x in A, p(x) is true.

      This statement is equivalent to the following.

      (b) (∃x ∈ A)(¬p(x))

      There is at least one x in A for which p(x) is false.

      Similarly, statements (c) and (d) are equivalent

      (c) ¬(∃x ∈ A)(p(x))

      It is false that there is at least one x in A for which p(x) is true.

      (d) (∀x ∈ A)(¬p(x))

      For all x in A, p(x) is false.

      Here is a method for negating quantified sentences. Starting at the beginning of the sentence, change each ∀ to ∃ and each ∃ to ∀. Then negate the proposition governed by the quantifier. In verbal sentences, the phrase “such that” is part of the quantifier “there exists.” In negating sentences, when we change “there exists” to “for all,” the phrase “such that” vanishes with “there exists.” When we change “for all” to “there exists,” the phrase “such that” appears with “there exists.”

       Examples (1.8)

       1. Statement: (∀x ∈ )(x ∈ )

      For each x ∈ , x ∈ .

      Negation: (∃x ∈ )(x ∉ )

      There exists x ∈ such that x ∉ .

      Remark. The symbol ∈ plays two different roles in the sentence (∀x ∈ )(x ∈ ). In the expression “∀x ∈ ,” the symbol ∈ describes the subject of the sentence. It says that the sentence is about any element x of .