While introducing formal logical notation, the book I'm reading says the following:
"$\forall x$ in $D$. $P(x)$" can be written as "$\forall x (x$ in $D \rightarrow P(x)$".
"$\exists x$ in $D$ such that $P(x)$" can be written as "$\exists x(x$ in $D \wedge P(x))$".
Why is it that the first conversion uses $\rightarrow$ while the second $\wedge$? The way I see, the symbol $\wedge$ can be used in both the statements and they should work.
Please explain.
Note that the first one can hold even if $D$ is empty, but for the second one a special $x$ must exist in D
Compare these two:
(a) Every leopard that lives in the north pole has spots on its skin.
(b) There exists a leopard that lives in the north pole and has spots on its skin.
It will be good to note what would happen if we use $\land$ in instead of $\to$, compare these two: $D={\text{Set of leopards living in the north pole}}$
$P(x) \iff x \text{ has spots on its skin}$
(a) Every $x$ is a leopard that lives in the north pole and has spots on its skin. ($\forall x,x \in D \land Spot(x)$)
(b) Every leopard that lives in the north pole has dots on its skin. ($\forall x,x\in D\to Spot(x))$)
