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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. $\exists d(\text{Rainy}(d) \wedge \text{~Cold}(d))$

Can I write logic for "All rainy days are cold." and then do the negation. Will that be correct?
Yes. That will be correct.
Thanks!

Not all rainy days are cold.

In other words it says $\text{Some rainy days are not cold"}$

Given statement is
$\neg \forall d[R(d)\to C(d)]$
$\equiv \neg \forall d[\neg R(d) \vee C(d)]$
$\equiv \exists d[R(d)\wedge \neg C(d)]$
Hence option (D) is correct.

1. No rainy days are cold
2. All non-rainy days are cold
3. Some non-rainy days are cold.
4. Some rainy days are not cold.

Option $D$.

Now (A): "all days are rainy days and they are not cold " is the correct translation.

The translation of option (C) should be,

(C) ∃d(~R(d)->C(d)) = ∃d(R(d) V C(d)) = (∃dR(d))  V (∃dC(d))  ="Some day are Rainy days or some days are Cold"

option a statement may be , " all the rainy day are cold "

if i am wrong correct me please

Try this way

NOT (all rainy days are cold)

~($\forall$d Rainy(d)->Cold(d))

~($\forall$ ~Rainy(d) $\vee$cold(d))

$\exists$d( Rainy (d) $\wedge$~Cold(d))

OPTION D

### 1 comment

Nicely explained

not all rainy days are cold : meaning "there are some rainy days which are cold" = "some days are rainy and not cold".

∃d{R(d)  ¬C(d)}

ans = option D