Question

Determine the truth ∀alue of the following statements if the uni∀erse of discourse of each ∀ariable is the set of real numbers. Enter your answer as Tor F. 1. ∃x∀y(xy = 0)

2. ∃x (x² = -1)
∃. ∃x∀y≠0(xy = 1)
4. ∀xSy((x + y = 2) ∧ (2x - y = 1))
5. ∀x #0∃y(xy = 1)
6. ∃x(x² = 2)
7. ∀x∃y(x2 = y)
8. ∀x∃y (x = y²)
9. ∃x∃y (x + y ≠ y + x)
10. ∀x∃zty(z = (x+y)/2)
11. ∃x∃y ((x + 2y = 2)^(2x + 4y = 5))
12. ∀x∃y(x + y = 1)

Answer 1

1. ∃x∀y(xy = 0) is false. 2. ∃x (x² = -1) is false. 3. ∃x∀y≠0(xy = 1) is true. 4. ∀xSy((x + y = 2) ∧ (2x - y = 1)) is true. 5. ∀x #0∃y(xy = 1) is true. 6. ∃x(x² = 2) is true. 7. ∀x∃y(x2 = y) is false. 8. ∀x∃y (x = y²) is false. 9. ∃x∃y (x + y ≠ y + x) is true. 10. ∀x∃zty(z = (x+y)/2) is true. 11. ∃x∃y ((x + 2y = 2)^(2x + 4y = 5)) is false. 12. ∀x∃y(x + y = 1) is false.

Step-by-step explanation:

1. ∃x∀y(xy = 0) - False
2. ∃x (x² = -1) - False
3. ∃x∀y≠0(xy = 1) - True
4. ∀xSy((x + y = 2) ∧ (2x - y = 1)) - True
5. ∀x #0∃y(xy = 1) - True
6. ∃x(x² = 2) - True
7. ∀x∃y(x2 = y) - False
8. ∀x∃y (x = y²) - False
9. ∃x∃y (x + y ≠ y + x) - True
10. ∀x∃zty(z = (x+y)/2) - True
11. ∃x∃y ((x + 2y = 2)^(2x + 4y = 5)) - False
12. ∀x∃y(x + y = 1) - False