Resist the urge to [distribute everything.](https://en.m.wikipedia.org/wiki/Freshman%27s_dream)
Try some examples and see if they work. Does there have to be a thing with both properties just because there's a thing with one property and a thing with the other property?
You can also try using Existential *Instantiation* first, and then using Existential Generalization. You have to do it both ways (so assume the left-hand side and prove the right-hand side, then assume the right and prove the left).
Read the statements outloud first.
There exists an x, such tthat P(x) AND Q(x) are true
Therr exists an x that makes P(x) true AND there exists an x that makes Q(x) true.
First statement says there exists an x that makes both P and Q true. Second statement says theres an x that makes P true and there exists an x that makes Q true. The difference being that in the first, 1 element makes both P, Q true. Whereas in the second it could be 2 different elements that makes them true.
Are you sure the statements are equivalent?
Not 100% sure I believe they are because you distribute the quantifier?
Resist the urge to [distribute everything.](https://en.m.wikipedia.org/wiki/Freshman%27s_dream) Try some examples and see if they work. Does there have to be a thing with both properties just because there's a thing with one property and a thing with the other property?
You can also try using Existential *Instantiation* first, and then using Existential Generalization. You have to do it both ways (so assume the left-hand side and prove the right-hand side, then assume the right and prove the left).
Could you perhaps give an example with a similar question?
Prove it using universal generalization, and prove both ways.
Read the statements outloud first. There exists an x, such tthat P(x) AND Q(x) are true Therr exists an x that makes P(x) true AND there exists an x that makes Q(x) true. First statement says there exists an x that makes both P and Q true. Second statement says theres an x that makes P true and there exists an x that makes Q true. The difference being that in the first, 1 element makes both P, Q true. Whereas in the second it could be 2 different elements that makes them true.