I'm currently in a boolean algebra class, and we are asked if the statement:
$$ \exists xM(x) \wedge \exists xD(x) $$
is a proposition. Although I know that it is a proposition, I was wondering if anyone knew if it could be simplified to
$$ \exists x (M(x) \wedge D(x)) $$
Since I know that this is true at least:
$$ \forall x(M(x) \wedge D(x)) \equiv \forall x M(x) \wedge \forall x D(x) $$
Perhaps let $M$ be "Man who likes cookies." Let $D$ be "Woman who likes dogs."
The first statement $$ \exists xM(x) \wedge \exists xD(x) $$ means that there is a man who likes cookies (Me) and there is a woman who likes dogs (My wife).
But, the second statement $$ \exists x(M(x) \wedge D(x)) $$ means "There exists a person that is both a man who likes cookies and a woman that likes dogs." Less likely. The two statements are not equivalent.