Question on Logical Expressions

What is the logical translation of the following statement?

                              “None of my friends are perfect.”

Answer : D

Explanation :

This question is not properly explained. Looks like F(X) returns TRUE if X is a friend and P(X) returns TRUE is X is prefect.

Analyzing Options Method

Option A :

This options says that, there exist a person X such that he is a friend to me AND he is not perfect. This sentence basically says :"Some of my friends are not perfect". So, obviously it is not saying “None of my friends are perfect.”.

Option B :

This options says, there is a person who is not a friend to me but he is perfect. Means "Some of people who are not friends with me are perfect". Obviously it does not say “None of my friends are perfect.”.

Option C :

This option says, there exist a person X who is not a friend to me and he is not perfect as well. Means "Some of people who are not friends with me are not perfect". Obviously it does not mean “None of my friends are perfect.”.

Option D :

This option basically says, there does not exist a person who is friends with me and perfect as well. That means "None of my friends are actually perfect". There could be persons who are friends with me but they are not perfect OR they are perfect but they are not friends with me. This is exactly we are looking for.

Derivation Approach

 “None of my friends are perfect.”

=> Vx (F(x) --> ~P(x)) [V means For All, ~ is negate operation]

=> Vx(~F(x) v ~P(x))

=> ~Ex(F(x) ^ P(x)) [E means There Exist]

Topics to revise :
Logical Expressions

References :