I am having trouble getting started with predicate logic proof. The task is to either prove that the statement is valid, or provide a valid example of where its false.
Here's the statement: $(\exists x)[R(x) \vee S(x)] \rightarrow (\exists x)R(x) \vee (\exists x)S(x)$
The intuition from looking at it, is that its equivalent, but having trouble with getting started. Is it as simple as:
$R(a) \vee S(a) \quad\text{[instantiate]}$
$(\exists x)R(a) \vee (\exists x)S(a) \quad\text{[generalize]}$
Anyone have any suggestions?
You have instantiated the existential statement into a disjunction $R(a)\vee S(a)$ and now wish to derive a disjunction of existential statements $(\exists x~R(x))\vee(\exists x~S(x))$.
Because you will want to eliminate the former disjunction, therefore a Proof by Cases is required.
$$\begin{array}{|l|}\hline\exists x~(R(x)\vee S(x))\\\hline R(a)\vee S(a)\\\quad\begin{array}{|l|}\hline R(a)\\\hline \vdots\phantom{\exists x~R(x)}\\(\exists x~R(x))\vee(\exists x~S(x))\\\hline\end{array}\\\quad\begin{array}{|l|}\hline S(a)\\\hline\vdots\phantom{\exists x~S(x)}\\(\exists x~R(x))\vee(\exists x~S(x))\\\hline\end{array}\\(\exists x~R(x))\vee(\exists x~S(x))\\\hline\end{array}$$
Now you just need to derive the later disjunction in each case. The how-to should be clear enough.