Solution to III. Determine the truth value of each of these statements if the domain consists of … - Sikademy
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Archangel Macsika

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

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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.

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