The CORRECT formula for the sentence, "not all rainy days are cold" is

\forall d (R a i n y (d) \land \sim C o l d (d))

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\forall d (\sim R a i n y (d) \to C o l d (d))

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\exists d (\sim R a i n y (d) \to C o l d (d))

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\exists d (R a i n y (d) \land \sim C o l d (d))

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Solution

The correct option is D $\exists d (R a i n y (d) \land \sim C o l d (d))$
Not ( all rainy days are cold):
$\sim (\forall d$ $R a i n y (d) \to C o l d (d)))$
$\equiv\sim (\forall d (\sim R a i n y (d) \lor C o l d (d)))$
$\equiv \exists d (R a i n y (d) \land \sim C o l d (d))$
Alternate Method:
Not all rainy days are cold is same as some rainy days are not cold which is same as
$\exists d (R a i n y (d) \land \sim C o l d (d))$

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Choose the option that completes the sentence correctly:

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