The propositional logic expression for ∃x∀yP(x,y)

Question

Where u.d. of x is {1,2,3} and y is {a,b}

The given answer is ((1,a)Λ(1,b)) V ((2,a)Λ(2,b)) V ((3,a)Λ(3,b))

But I get the expression ((1,a)V(2,a)V(3,a)) Λ ((1,b)V(2,b)V(3,b))

Why is my one wrong and the given answer correct?

Answer 1

Your answer corresponds to the expression $\forall y\exists xP(x,y)$.

Answer 2

The quantifier closest to the predicate maps to the innermost connective.

The quantifier farthest from the predicate maps to the outermost connective.

$\begin{align} \color{red}{∃x}\;\color{blue}{∀y}\;P(x,y) & = \color{red}{\exists x}\;\big(P(x,a)\color{blue}{\wedge} P(x,b)\big) \\ & = \big(P(1,a)\color{blue}{\wedge} P(1,b)\big)\color{red}{\vee} \big(P(2,a)\color{blue}{\wedge} P(2,b)\big)\color{red}{\vee} \big(P(3,a)\color{blue}{\wedge} P(3,b)\big) \\[3ex] \color{blue}{\forall y}\;\color{red}{\exists x}\;P(x,y) & = \color{blue}{\forall y}\;\big(P(1,y)\color{red}{\vee} P(2,y)\color{red}{\vee} P(3,y)\big) \\ & = \big(P(1,a)\color{red}{\vee} P(2,a)\color{red}{\vee} P(3,a)\big) \color{blue}{\wedge} \big(P(1,b)\color{red}{\vee} P(2,b)\color{red}{\vee} P(3,b)\big) \end{align}$