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The CORRECT formula for the sentence, "not all Rainy days are Cold" is

  1. $\forall d (\text{Rainy}(d) \wedge \text{~Cold}(d))$
  2. $\forall d ( \text{~Rainy}(d) \to \text{Cold}(d))$
  3. $\exists d(\text{~Rainy}(d) \to \text{Cold}(d))$
  4. $\exists d(\text{Rainy}(d) \wedge \text{~Cold}(d))$
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3 votes
3 votes
"all Rainy days are Cold" : ∀d(Rainy(d)->Cold(d))
"not all Rainy days are Cold" : ~∀d(Rainy(d)->Cold(d))
                                        <=>∃d~(Rainy(d)->Cold(d))
                                        <=>∃d~(~Rainy(d)VCold(d))
                                        <=>∃d(Rainy(d)∧~Cold(d))

so Ans D is correct
1 votes
1 votes

(A)∀d(R(d)⋀~C(d)) = d(~(~R(d) V C(d))) (taking negation common)

                                   =∀d(~(R(d)->C(d)))= All days are not Rainy days and also are not Cold

(B)d(~R(d)->C(d))=The day which are not Rainy day are Cold

(C)∃d(~R(d)->C(d))=∃d(R(d)VC(d))=Some day are Rainy days or some days are Cold

(D)∃d(R(d)⋀~C(d))= Some Rainy days are not Cold

                                = ~ (∀d(R(d)->C(d))) (taking negation common)

                                =not all Rainy days are Cold

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1 votes
1 votes

(A) Note that (p ∧ ~q) ≡ ~(p -> q). So it means rainy day to cold implication is false for all days. Which means non-rainy days are cold. (B) For all days, if day is not rainy, then it is cold [Non-Rainy days are cold] (C) There exist some days for which not rainy implies cold. [Some non-rainy days are cold] (D) Note that (p ∧ ~q) ≡ ~(p -> q). So it means rainy day to cold implication is false for some days. Which means not all rainy days are cold.

1 votes
1 votes
"Not all rainy days are cold."

Which means..

There is a rainy day which is not cold.

Which is equivalent to ∃d(Rainy(d)∧~Cold(d))

(as restriction of an existential quantification is same as existential quantification of a conjunction.)

so option D is correct.
Answer:

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