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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

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  • $\begingroup$ Both are correct. $\forall$ needs $\to$ while $\exists$ needs $\land$. $\endgroup$ Jun 4, 2019 at 15:31
  • $\begingroup$ See similar post. $\endgroup$ Jun 4, 2019 at 15:33

3 Answers 3

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Welcome to Stack Exchange!

Your translations are correct. Here's a simple rule for translating phrases like "some clear explanations" and "no excuses":

  • The sentence "some A's are B's" translates as $\exists x (A(x) \land B(x))$. (Some things are both A's and B's.)
  • The sentence "all A's are B's" translates as $\forall x (A(x) \implies B(x))$. (Everything, if it is an A, is also a B. In other words, everything is either not-an-A, or a B.)

So, "Some clear explanations are satisfactory" translates as $\exists x (P(x) \land Q(x))$.

How about "No excuses are clear explanations"? Well, we can rephrase that as "All excuses are things that are not clear explanations". So this is $\forall x (R(x) \implies \neg P(x))$.

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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 $\forall x$.

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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 :)

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