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Translate each of these statements into logical expressions
using predicates, quantifiers, and logical connectives.
a) No one is perfect.
b) Not everyone is perfect.
c) All your friends are perfect.
d) At least one of your friends is perfect.

4 Answers

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a) No one is perfect. == Not ( one is perfect) = ~ (∃x(px))= ∀x ~p(x)= Every one is imperfect.
b) Not everyone is perfect.== Not (everyone is perfect.)= ~( ∀x(px))=∃x ~p(x)= Atleast one is imperfect.
c) All your friends are perfect. == if there is a person who is your friend then he is perfect== ∀x( F(x)→P(x))
d) At least one of your friends is perfect. == There is a person who is your friend who is perfect.

∃x (F(x)∧P(x))

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plz correct me if wrong!!

P(x) : perfect

F(x) :friends

(a)∽∃x(P(x))

(b)∽∀x(P(x))

(c)∀x(F(x)------>P(x))

(d)  i am thinking in this way 

ALL FRIENDS ARE PERFECT - NO FRIEND IS PERFECT

 how to write above sentence...?? in logic

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Equivalent answers,

$P(x):$ X is Perfect
$F(x):$ X is a friend of mine

(a) $\forall x \neg P(x)$
(b) $\exists x \neg P(x)$
(c) $\forall x,  F(x) \rightarrow P(x)$
(d) $\exists y \forall x F(x) \wedge P(y)$
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P(x) : perfect

F(x) :friend

a) ∽∃x(P(x))

B)  ∽∀x(P(x))

C) ∀x(F(x)->P(x))

D) ∃x(Friend(x) ^ Perfect(x))

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