III. Determine the truth value of each of these statements if the domain consists of all integers. State your reason. 1. ∀𝑥, (𝑥 2 > 𝑥) 2. ∃𝑦, (𝑦 < 𝑦 2 − 1) 3. ∀𝑦, (𝑦 2 ≠ 𝑦) 4. ∃𝑥, 𝑦, (4𝑥 > 5𝑦) where 𝑥 < 𝑦 5. ∀𝑥, 𝑦, (𝑥𝑦 > 0) where 𝑥 = y

III. Let us determine the truth value of each of these statements if the domain consists of all integers.

1. Since it is not true that $0^2=0>0,$ we conclude the that truth value of $∀𝑥 (𝑥^2 > 𝑥)$ is false.

2. Since for $y=-1$ we get that $-1<0=(-1)^2-1,$ we conclude that the truth value of $∃𝑦 (𝑦 < 𝑦^2 − 1)$ is true.

3. Since it is not true that $0^2\ne0,$ we conclude that the truth value of $∀𝑦 (𝑦^ 2 ≠ 𝑦)$ is false.

4. Since for $x=-10, \ y=-9$ we have that $x<y$ and $4x=-40>-45=5y,$ we conclude that the truth value of $∃𝑥, 𝑦 (4𝑥 > 5𝑦)$ where $𝑥 < 𝑦$, is true.

5. Since for $x=y=0$ it is not true that $0\cdot 0>0$ we get that the truth value $∀𝑥, 𝑦, (𝑥𝑦 > 0)$ where $𝑥 = y$, is false.