recall that ε is pronounced “epsilon.”
1. (∃x ∈
2. (∃x ∈
3. (∀x ∈
4. (∀x ∈
5. (∀x ∈
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
14. For each integer x, if x ≤ 0 then —x ≥ 0.
15. For each x ∈
16. For each x ∈
17. For each x ∈
18. For all real numbers x and y, if x ≤ y and x ≥ y then x = y.
19. For each x ∈
20. There exists a negative real number r such that
21. There exists x ∈
22. There exists x ∈
23. There exists x ∈
24. For each real number x such that x ≠ 1. if
Remark. The preceding sentence may be written, “For each real number x, if x ≠ 1 then if
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 ∈
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 ∈