Consider the following formula and its two interpretations $I_1$ and $I_2$.

$\alpha: (\forall x)\left[P_x \Leftrightarrow (\forall y)\left[Q_{xy} \Leftrightarrow \neg Q_{yy} \right]\right] \Rightarrow (\forall x)\left[\neg P_x\right]$

$I_1$ : Domain: the set of natural numbers

$P_x$ = 'x is a prime number' $Q_{xy}$ = 'y divides x' $I_2$ : same as $I_1$ except that $P_x$ = 'x is a composite number'.

Which of the following statements is true?

$\alpha: (\forall x)\left[P_x \Leftrightarrow (\forall y)\left[Q_{xy} \Leftrightarrow \neg Q_{yy} \right]\right] \Rightarrow (\forall x)\left[\neg P_x\right]$ I₁: domain the set of natural numbers Px = x is a prime number = 2, 3, 5, 7, 11, 13… Qxy = y divides x = means y is also prime number = 2,3,5,7,11,13, I₂: px = x is composite number (i.e not prime number) = 4,6,8,9,10,12,14,….. Qxy = 2,3,4,5,7,……. = mean all the numbers from expression, we know I₁ satisfies $\alpha$,I₂ does not