Translating English statements into quantifiers and predicates. Consider the following statement: "All your friends are perfect." This can be translated into quantifiers and predicates as follows :
Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people. So the statement can be written as : ∀x (F (x) → P (x)).
However, consider the statement: At least one of your friends is perfect. The answer is  ∃x (F (x) ∧ P (x)), but why it is not ∃x (F (x) → P (x)) ?
 A: In the first case we need an expresion that resume, if x is your friend then x is perfect, becuase all your friend are perfect and in the case of x isn´t your friend we dont know if is perfect or not but it doesn´t affect the steament so you have two options F(x) = False or True,the firts one is x is not your friend and we have to remember that False$\rightarrow$ q $\equiv$ True, for all q, so that case doesent affect the value of our steament, then when $F(x)\equiv$ True, True$\rightarrow$ P(x) $\equiv$ True if and only if P(x)$\equiv$ True, then we can notice the steament does exactly what we want, doest say anithing about people that are not friends , and comnfirms that if x is your friend is perfect.
I think that in the second case you are missing and $\equiv$ True because that will mean that there is a person that is your friend and it is perfect at the same time, meaning at least one of your friends are perfect, and we can't use an implies becuase we only know that at least one of your friends is perfect so we could have the case of a friend who isnt perfect that will mean F(x)$\equiv$ True and P(x)$\equiv$ False and we know that True$\rightarrow$ False $\equiv$ False, and the explesión $\exists x$ F(x)$\rightarrow$ P(x) can be true and not confirm that you have at least a perfect friend because if x is not your friend , the expresion is always true.
for sintesis if you use F(x)$\rightarrow$ P(x), it could not be always true, and even if it is true it could not mean what you intended to say.
A: The last expression can be translated to English as "There is at least one person that if he is your friend then he is perfect". So there is the possibility that you don't have friends. The first expression requires you to have at least one friend. $F\land T=F$, while $F\to T=T$. Similarly, $F\land F=F$, while $F\to F=T$.
