Question: Given: 

* P(x) = "x is a clear explanation"
* Q(x) = "x is satisfactory"
* R(x) = "x is an excuse

I need to translate 

* a) Some clear explanations are satisfactory.
* b) No excuses are clear explanations. (All excuses are not clear explanations)

I have a lot of difficult distinguishing when to use a conjunction and when to use an implication. Here are the translations I came up with:

* a) ∃x(P(x)∧Q(x))
* b) ∀x(R(x) --> ¬P(x))

Are these correct? Regardless of my correctness, can you provide an explanation why I was right/wrong in using the implication over the conjunction and vice versa. I would like to get their correct uses straight in my head.

Thanks

Answer: 
Your answers for both parts are correct.

For part a), "some clear explanations are satisfactory" means that "there exists a clear explanation which is satisfactory", which requires $x$ to be both a clear explanation ($P(x)$) and satisfactory ($Q(x)$). And "some" indicates $\exists x$.

For part b), "no excuses are clear explanations" means that "if we have an excuse, then it is not a clear explanation", which means for all $x$, $x$ becoming an excuse ($R(x)$) implies it is not a clear explanation ($\lnot P(x)$). And "no" indicates $\forall x$.
Your translations are correct.

I understand the difference between conjunction and implication can be confusing at first. Just remember almost all translations with the existential claims (∃x) use conjunction, most universal claims (∀x) use implication.

∃x(P(x)∧Q(x))  

translates to:  

"There exists a clear explanation, and it is satisfactory."  

But  

∃x(P(x)-->Q(x)) which we know is equivalent to
∃x(~P(x) V Q(x))  

would translate to:  

"There exists a (member of the universe) such that if it were a clear explanation, it would be satisfactory."
This sentence basically means there may or may not be a clear explanation that is satisfactory, which is a very weak claim, so it is kinda rare to see implication being used with existential claims.   

also:   

∀x(R(x) --> ¬P(x))  

translates to:  

"All excuses are not clear explanations."  

Try to think of it like an if-then statement. If there is an excuse, then it is not a clear explanation.  

but if the formula was:  

∀x(R(x) ∧ ¬P(x))  

It would mean every (member in the universe) is an excuse, and not a clear explanation.  

So basically you'll use ∧ with ∃, and --> with ∀ like 99% of the time. If you have not yet learned about defining a universe, or universal set, then you will probably use use ∧ with ∃, and --> with ∀ 100% of the time, because statements like ∀x(R(x) ∧ ¬P(x)) and ∃x(P(x)-->Q(x)) wouldn't make sense.

Hope this helped. Feel free to ask any other questions, I just took discrete math last semester :)