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

as @vintl mentioned the reason, I too also feel the answer should be (c)

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https://gateoverflow.in/1642/gate1998-1-5?show=295562#c295562

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q only if p $=$ if no p then no q $=$ ~p → ~q $=$ q → p

Answer 1

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 2

option  A. is correct answer

Answer 3

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             

 

Answer 4

1. a and b are the same size if a = b
a = b → SameSize(a, b)
2. a and b are the same size only if a = b
SameSize(a, b) → a = b

Try to map Option A to above, simpler, example.

Ref: https://faculty.washington.edu/smcohen/120/Chapter7.pdf