Эротические рассказы
Эротические рассказы

Write Your Own Proofs. Amy BabichЧитать онлайн книгу.

Write Your Own Proofs - Amy Babich

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to the set image.

       2. Statement: (∀x ∈ image)(∃y ∈ image)(x + y = 0)

      For each x ∃ image, there exists y ∈ image such that x + y = 0.

      Negation: (∃x ∈ image)(∀y ∈ image)(x + y ≠ 0)

      There exists x ∈ image such that for all y ∈ image, x + y ≠ 0.

       3. Statement: (∀x ∈ image)(∀y ∈ image)((x > y) ⟶ (∃z ∈ image)(x > z > y))

      For each x ∈ image, for each y ∈ image, if x > y then there exists z ∈ image such that x > z > y.

      Negation: (∃x ∈ image)(∃y ∈ image)((x > y) ∧ (∀z ∈ image)((x ≤ z) ∨ (z ≤ y))) There exist x ∈ image, y ∈ image such that x > y and for all z ∈ image, x ≤ z or z ≤ y.

      Remark. The third example is more complicated than the other two. Hence we offer a step-by-step analysis of the process of negation.

      In Example 3, we negate the following statement.

      (∀x ∈ image)(∀y ∈ image)((x > y) ⟶ (∃z ∈ image)(x > z > y))

      That is, we produce a statement that is logically equivalent to the following.

      ¬((∀x ∈ image)(∀y ∈ image)((x > y) ⟶ (∃z ∈ image)(x > z > y)))

      For our first step, we change the quantifiers at the beginning of the sentence and move the symbol ¬ to their right.

      (∃x ∈ image)(∃y ∈ image)(¬((x > y) ⟶ (∃z ∈ image)(x > z > y)))

      Notice that the statement governed by the two initial quantifiers has the form ¬(p ⟶ q). Since ¬(p ⟶ q) is logically equivalent to p ∧ ¬q, we obtain the following sentence.

      (∃x ∈ image)(∃y ∈ image)((x > y) ∧ ¬(∃z ∈ image)(x > z > y))

      Now we transform the sentence ¬(∃z ∈image)(x > z > y) by changing the quantifier and moving ¬ to the right.

      (∃x ∈ image)(∃y ∈ image)((x > y) ∧ (∀z ∈ image)(¬(x > z > y)))

      Since x > z > y is shorthand for (x > z) ∧ (z > y), we have the following sentence.

      (∃x ∈ image)(∃y ∈ image)((x > y) ∧ (∀z ∈ image)(¬((x > z) ∧ (z > y))))

      Since ¬(p ∧ q) is logically equivalent to ¬p ∧ ¬q, we transform the sentence as follows.

      (∃x ∈ image)(∃y ∈ image)((x > y) ∧ (∀z ∈ image)(¬(x > z) ∨ ¬(z > y)))

      Finally, since ¬(a > b) can be written more simply as a ≤ b, we get the sentence below.

      (∃x ∈ image)(∃y ∈ image)((x > y) ∧ (∀z ∈ image)((x ≤ z) ∨ (z ≤ y)))

      Remark. Every step in simplifying a negation moves the symbol ¬ further to the right. When all the ¬ symbols are as far to the right as possible, we have simplified the negation as much as possible.

      Exercises (1.7) Classify each

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