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What is the logical translation of the following statement?

"None of my friends are perfect."

1. $∃x(F (x)∧ ¬P(x))$
2. $∃ x(¬ F (x)∧ P(x))$
3. $∃x(¬F (x)∧¬P(x))$
4. $¬∃ x(F (x)∧ P(x))$

A) "Some of my friends are not perfect"

B) "Some are not my friends and are perfect"

C) "Some are not my friends and are not perfect"

D) "There exist no one who is my friend and is perfect"
can we solve it using truth table?
Yes but it'll take lot of time

1. some of my friends are not perfect
2. some of those who are not my friends are perfect
3. some of those who are not my friends are not perfect
4. NOT (some of my friends are perfect) / none of my friends are perfect

Correct Answer: $D$

edited

@ arjun sir,

Can we retransform the given statement like this...."None of my friends are perfect" to "All my freinds are not perfect"  so that we can directly write like this, for  x( F(X)-->~P(X)) ....

What do you say?

None of my friends are perfect can be written an "all of my friends are not perfect".

Now all of my friends are not perfect could be written as:

$\forall x (F(x) \implies \lnot P(x))$

$\forall x (\lnot F(x) \cup \lnot P(x))$

By de Morgan's law: -

$\lnot \exists(F(x) \cap P(x))$

Which is equivalent to option D.

Sir, what is the right way to interpret it?

We can also do this:  "Atleast one of my is perfect" $\exists x(F(x) \wedge P(x) )$ and Negation of this will be $\neg \exists x(F(x) \wedge P(x)) = \forall x\neg(F(x)\wedge P(x))= \forall x (\neg F(x) \lor \neg P(x))$
• $F\left(x\right): x \text{ is my friend.}$
• $P\left(x\right):\text{x is perfect.}$

$\text{“None of my friends are perfect"}$ can be written like

$\forall x[F(x)\implies\neg P(x)]$
$\equiv \forall x[\neg F(x)\vee \neg P(x)]$
$\equiv \forall x\neg[F(x)\wedge p(x)]$
$\equiv \neg\exists x[F(x)\wedge P(x)]$

by

@vnc

Can u elaborate after 2nd last step?
why you again negate theirexist?????in last line???

plz tell me my approach is right or not

for all people in the world who is my friend then not perfect i.e ∀x[F(x)--->~P(x)]

=∀x[~F(x)∨~P(x)]
=∀x~[F(x)∧p(x)] now according to me last line should be...                                                                      =∃x[F(x)∧P(x)] but why you put negation??????
Wrong calculation ... correct it ...
"None of my friends are perfect."

It is NOT the complement of "All of my friends are perfect"   So A is not the answer. (A frequently done mistake)

It is the complement of "At least one of my friend is perfect"  So D is the answer.
by

If I take complement of "All of my friends are perfect" then what will it mean? What will be the semantic meaning of it.

@ Abhishek Gupta 1
"At  least one of my friends is not perfect."

"None of my friends are perfect."

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