GATE CSE
First time here? Checkout the FAQ!
x
+9 votes
795 views

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. $\exists d(\text{Rainy}(d) \wedge \text{~Cold}(d))$
asked in Mathematical Logic by Veteran (87.2k points)  
edited by | 795 views
Is there typo in last option ? please correct it !
corrected.

You Choose UOD(Universe Of Discourse) Correctly , you can answer easily

9 Answers

+13 votes
Best answer

Not all rainy days are cold.

In other words it says "Some rainy days are cold" or "Some rainy days are not cold"
 

Given statement is
~Vd[R(d)->C(d)]
<=>~Vd[~R(d)VC(d)]

<=> ∃ d[R(d) ^ ~C(d)]
D)

 

answered by Loyal (3.7k points)  
selected by
+16 votes
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
answered by Loyal (2.8k points)  
Is option (A) statement correct?
Statement A  shoud be "all days are rainy days and they are not cold "
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"

+5 votes
Try this way

NOT (all rainy days are cold)

~(¥ d Rainy(d)->Cold(d))

~(¥d ~Rainy(d) DISJUNCTION cold(d))

∃d( Rainy (d) CONJUNCTION ~Cold(d))

OPTION D
answered by Loyal (3.4k points)  
Nicely explained
+2 votes

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) \scriptstyle \wedge ¬C(d)}

 

ans = option D

answered by Veteran (28.7k points)  
0 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

answered by Veteran (58.3k points)  
edited by
Is option (A)  Translation correct?

It should be "all days are rainy days and they are not cold ".
0 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.

answered by Loyal (4k points)  
0 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.
answered by Active (1.6k points)  
0 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
answered by Active (1.2k points)  
0 votes

Option d bcz it means there is some days which are rainy not cold 

answered by Active (2.1k points)  


Top Users Sep 2017
  1. Habibkhan

    6504 Points

  2. Arjun

    2254 Points

  3. Warrior

    2234 Points

  4. nikunj

    1980 Points

  5. manu00x

    1726 Points

  6. Bikram

    1726 Points

  7. SiddharthMahapatra

    1718 Points

  8. makhdoom ghaya

    1680 Points

  9. A_i_$_h

    1668 Points

  10. rishu_darkshadow

    1554 Points


25,999 questions
33,568 answers
79,491 comments
31,035 users