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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

  1. $(\neg p \wedge r) \wedge (\neg r \rightarrow (p \wedge q))$
  2. $(\neg p \wedge r) \wedge ((p \wedge q) \rightarrow  \neg r)$
  3. $(\neg p \wedge r) \vee ((p \wedge q) \rightarrow  \neg r)$
  4. $(\neg p \wedge r) \vee (r \rightarrow (p \wedge q))$
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10 Answers

Best answer
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60 votes
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.
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32 votes

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))

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

Answer is A
Answer:

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