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

Answer: option A

Pre-requisite:

1. When  $P \Rightarrow Q$ is given, it means condition P is sufficient for condition Q.
    
2. When the first condition is necessary for the second condition, then the second condition is sufficient for the first condition. i.e If Q is necessary for P then P is sufficient for Q.
    
3. Precedence 
   

 

 

 

 

 

 

 

 

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 as

 

 Solution:

We will solve the question in parts

Part 1: “It is not raining and it is pleasant”  

• It is not raining =  $\sim p$
• and = $\wedge$
• it is pleasant = r

so finally it is equivalent to $\sim p \wedge r$

 

Part 2: “It is not pleasant only if it is raining and it is cold”   

In part 2, remember that $\wedge$  has more priority than $\Rightarrow$ and hence “it is raining and it is cold”  will be combined first. - (By pre-requisite 3)

It is not pleasant = $\sim r$

it is raining and it is cold = $p \wedge q$

only if = $\Rightarrow$

so this is equivalent to $\sim r \Rightarrow p \wedge q $

Finally both the parts are connected using $\wedge$ operation.

$(¬p∧r)∧(¬r→(p∧q))$

 

 

Direction of Implication $\Rightarrow$ ?

“It is not pleasant only if it is raining and it is cold”  
 

 

 “It is not pleasant” only when  “It is raining and it is cold”, this tells,  you will not feel pleasant only when it is raining and it is cold. So “raining and cold” is a necessary condition to “not feel pleasant”. Hence “not feel pleasant”  is a sufficient condition for “raining and cold”  – (By pre-requisite 2)
 

Now since “not feel pleasant”  is a sufficient condition for “raining and cold”, we can write $\left (\sim r\rightarrow \left ( p \wedge q \right ) \right )$ – (By pre-requisite 1)


PC: https://web.stanford.edu/class/archive/cs/cs103/cs103.1152/lectures/07/Slides07.pdf

Answer 2

Answer given is (A) in Answer key .

But Answer according to me is (C)  :

Explanation :

As the day cannot be pleasant and not pleasant at the same time.

Proof :

 p :  it is raining

 q:  it is cold

 r :  it is pleasant

Statement : “ It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold . “

According to this statement : Draw Venn Diagram

Therefore Answer is (C) :

 

Proof 2 :

Also (A) when solved yields to p^c /\ r  (shown below ) :

Statement: “It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold. “

Now “ not pleasant” is coming first in statement and then “ it is raining and it is cold “

But is representation it is :

Similar is the case in Gate CS 2006 (full question with explanation is given in page 2)

Statement: “Tigers and lions attack if they are hungry or threatened.”

Attack comes first and then “hungry or threatened “but it is represented as

{(hungry (x) v threatened (x)) -> attacks (x)} in

Similar questions from previous year gate exam are given below :

In GATE CS 2009

Question

Which one of the following is the most appropriate logical formula to represent the statement? "Gold and silver ornaments are precious". The following notations are used: G(x): x is a gold ornament S(x): x is a silver ornament P(x): x is precious

A	∀x(P(x)→(G(x)∧S(x)))
B	∀x((G(x)∧S(x))→P(x))
C	∃x((G(x)∧S(x))→P(x)
D	∀x((G(x)∨S(x))→P(x))

Answer : (D) 

Explanation :

=> This statement can be expressed as => For all X, x can be either gold or silver then the ornament X is precious => For all X, (G(X) v S(x)) => P(X). As X can’t be gold and silver at the same time.

In GATE-CS -2006

Question:

Which one of the first order predicate calculus statements given below correctly express the following English statement?

Tigers and lions attack if they are hungry or threatened. 

Answer: D

 Explanation :

The statement "Tigers and lions attack if they are hungry or threatened" means that if an animal is either tiger or lion, then if it is hungry or threatened, it will attack. So option (D) is correct. "and" doesn't mean that we will write "tiger(x) ∧ lion(x) ", because that would have meant that an animal is both tiger and lion, which is not what we want.

Also attack comes first in statement “Tigers and lions attack if they are hungry or threatened. “ but in representation answer is :  D i.e

{(hungry (x) v threatened (x)) -> attacks (x)}