Suppose that the universe of discourse of the atomic formula $P(x,y)$ is $\{1, 2, 3\}$. Write out the following propositions using disjunctions and conjunctions.
a) $\exists x \forall y \ P(x,y)$
Because there exists at least one $x$ for all $y$, my awnser is: $$\bigl(P(1,1) \lor P(1,2) \lor P(1,3)\bigr) \land \bigl(P(2,1) \lor P(2,2) \lor P(2,3)\bigr) \land \bigl(P(3,1) \lor P(3,2) \lor P(3,3)\bigr).$$
However my awnser doesn't seems to be right.
$\endgroup$ 21 Answer
$\begingroup$An universal quantifier is conjunctive over the domain, where as an existential quantifier is disjunctive.
If something is true for all entities, then it's true for this and that and every other entity.
If something is true for some entity, then it's true for this or that or some other entity.
$$\exists x~\forall y~P(x,y) {~=~ \bigvee_{x\in\mathcal D}\bigwedge_{y\in\mathcal D}P(x,y) \\~=~ (\bigwedge_{y\in\mathcal D}P(1,y))\vee(\bigwedge_{y\in\mathcal D}P(2,y))\vee(\bigwedge_{y\in\mathcal D}P(3,y))\\~=~ (P(1,1)\wedge P(1,2)\wedge P(1,3))\vee(\bigwedge_{y\in\mathcal D}P(2,y))\vee(\bigwedge_{y\in\mathcal D}P(3,y))\\~~\vdots}$$
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