Write Your Own Proofs. Amy Babich

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> 2) ⟶ (x > 3))

       5. (∀x ∈ )((∃k ∈ )(x = 2k) ⟶ (∃m ∈ )(x = 4m))

       6. For every natural number x, there exists a natural number y such that x ≠ y and for every natural number z, if x > z then y > z.

       7. For all natural numbers x and y, if x < y then there exists a natural number 2 such that x < z and z < y.

       8. For every integer x, either x is a natural number or –x is a natural number.

       9. For every natural number x, there exists a natural number y such that y² = x.

      10. For all integers x and y, if x – y is a natural number, then x > y.

      11. For each integer x, if |x – 2| > 6 then x > 8 or x < –4.

      12. There exists a natural number x such that for every integer y, x – y is a natural number.

      13. Given any positive real number ε, there exists a natural number k such that whenever n is a natural number such that n > k.

      14. For each integer x, if x ≤ 0 then —x ≥ 0.

      15. For each x ∈ , there exists y ∈ such that y = 2x.

      16. For each x ∈ , there exists y ∈ such that x = 2y.

      17. For each x ∈ , if x ≠ 0 then there exists y ∈ such that xy = 1 and for all z ∈ , if xz = 1 then y = z.

      18. For all real numbers x and y, if x ≤ y and x ≥ y then x = y.

      19. For each x ∈ , if there exists y ∈ such that y² = x, then y ∈ .

      20. There exists a negative real number r such that is real.

      21. There exists x ∈ such that for all y ∈ , x ≠ 2y and x ≠ 2y – 1.

      22. There exists x ∈ such that for all y ∈ , if there exists k ∈ such that yk = x then y = x or y = 1.

      23. There exists x ∈ such that for all y ∈ , x ≤ y.

      24. For each real number x such that x ≠ 1. if then x > 1.

      Remark. The preceding sentence may be written, “For each real number x, if x ≠ 1 then if then x > 1.” The phrase such that has been used with the universal quantifier to avoid the repetition of if . . . then. (It is also possible to write, “For each real number x, if x ≠ 1 then implies x > 1.”)

      25. There exists a real number x such that x is a rational number and x is not a natural number.

      26. For every natural number n, if n ≠ 1 then

      27. There exists x ∈ such that for all y ∈ , y² ≠ x.

      28. For each rational number x, the number x² is also a rational niunber.

      29. For each rational number x, there exists a rational number y such that y² = x.

      30. For all x ∈ , if there exists y ∈ such that y² = x, then for all z ∈ , z³ ≠ x.

      31. For all x ∈ , there exists y ∈ such that for all z ∈ ,

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