GATE CSE 2014 Set 3 | Question: 53

Question

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

• A. $\forall d (\text{Rainy}(d) \wedge \text{~Cold}(d))$
• B. $\forall d ( \text{~Rainy}(d) \to \text{Cold}(d))$
• C. $\exists d(\text{~Rainy}(d) \to \text{Cold}(d))$
• D. $\exists d(\text{Rainy}(d) \wedge \text{~Cold}(d))$

Comments

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

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Yes. That will be correct.

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Thanks!

Answer 1

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.

Answer 2

• A. No rainy days are cold
• B. All non-rainy days are cold
• C. Some non-rainy days are cold.
• D. Some rainy days are not cold.

Option $D$.

Comments

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

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 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"

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

 

if i am wrong correct me please

Answer 3

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

Comments

Nicely explained

Answer 4

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