GATE CSE 2017 | Set 2 | Question: 11

Question

Let $p, q, r$ denote the statements ”It is raining”, “It is cold”, and “It is pleasant”, respectively. Then the statement “It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold” is represented by

• A. $(\neg p \wedge r) \wedge (\neg r \rightarrow (p \wedge q))$
• B. $(\neg p \wedge r) \wedge ((p \wedge q) \rightarrow  \neg r)$
• C. $(\neg p \wedge r) \vee ((p \wedge q) \rightarrow  \neg r)$
• D. $(\neg p \wedge r) \vee (r \rightarrow (p \wedge q))$

Comments

p = it is raining 

q = it is cold

r = it is plesent

given statement-

"It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold"

jus obserbe this part - "It is not raining and it is pleasant, and "   == (¬p∧r)∧

option C,D is gone

then "it is not pleasant only if it is raining and it is cold" == ( ¬r→(p∧q) )

Hence option A is correct

 

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Option A and Option B are logically same ,both simplify to (p_negation intersection r) yet only A is correct? so for translation we would tick only that option which preserves the english structure?

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You don't simplify for translation-type questions. Here you pick the option that matches structurally.

Answer 1

1. "It is not raining and it is pleasant" can be written as $(¬p∧r)$

2. Now, "it is not pleasant only if it is raining and it is cold" is represented by $¬r\implies (p∧q)$  but $(p∧q) \not\implies ¬r $. Why? Because if it is not pleasant then we can conclude it must be raining and it is cold. However, it is raining and cold does not assure that it will be unpleasant. i.e., $p$ only if $q$ can be written as if $p$ then $q$ (not double implication).

So, ANDing clause $1.$ and $2.$ we get $(¬p∧r)∧(¬r→(p∧q))$

option A is correct.

Comments

@MiNiPanda

Yes bro ..I found that ..that y I hidden that comment :)

simply 

[ p sufficent for q ] , same as ,  [ q if p ]  ,same as,  [ if p then q ] ,  [p==>q]

[ p necessary for q ] , same as ,  [ q only if p ]  ,same as,  [ if ~p then ~q ] ,  [q==>p]

Now ok bro ?

 

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Can we think like this? :

¬r⟹(p∧q)

Here  ¬r  is false , so false implies anything is always true. Hence above expression will be TRUE.

But (p∧q)⟹¬r  in this expression  (p∧q) is TRUE and ¬r is FALSE , so it will be FALSE.Hence it is wrong.

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@minipanda what an explaination sister thanks

Answer 2

it is not pleasant only if it is raining and it is cold

it is not pleasant ---> ¬r

t is raining and it is cold----> (p∧ q)

therefore it becomes  (¬r→(p∧q))

Comments

thank you for this :)

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helpful

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thnaks man

 

Answer 3

$(\sim p \wedge r) \wedge (\sim r \rightarrow (p \wedge q))$

Answer is A

Answer 4

option A seems to be correct.

Answer 5

Translation of each formula into english statements :

(p’ $\wedge$ r) :- (not p) and r :- It is not raining and it is pleasant.

r’ → (p $\wedge$ q) :- (If not r, then p and q) $\equiv$ (not r only if p and q) $\equiv$ It is not pleasant only if it is raining and it is cold.

(p $\wedge$ q) → r’ :- (If p and q, then not r) $\equiv$ (p and q only if not r) $\equiv$ It is raining and it is cold only if it is not pleasant.

r → (p $\wedge$ q) :- (If r, then p and q) $\equiv$ (r only if p and q) $\equiv$ It is pleasant only if it is raining and it is cold.

• A. “It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold”
• B. “It is not raining and it is pleasant, and it is raining and it is cold only if it is not pleasant”
• C. “Either It is not raining and it is pleasant, or it is raining and it is cold only if it is not pleasant”
• D. “Either It is not raining and it is pleasant, or it is pleasant only if it is raining and it is cold”

Answer 6

p = ”It is raining”

q =  “It is cold”

r = “It is pleasant”

It is not raining and it is pleasant = ~p ∧ r

x only if y can be written as x→y

 it is not pleasant only if it is raining and it is cold) = ~r→(p ∧ q)

“(It is not raining and it is pleasant), and it is not pleasant only if (it is raining and it is cold)” =   (¬p∧r)∧(¬r→(p∧q))             

So Option A             

 

Comments

this is better than directly stating the right option. thanks!