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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*→

“(*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 **

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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

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Answer:

Pre-requisite:

- When $P \Rightarrow Q$ is given, it means condition P is
**sufficient**for condition Q.

- 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.

- Precedence

Let p,q,r denote the statements ”

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:** “*I**t 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)*

*I**t 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.

$(¬p∧r)∧(¬r→(p∧q))$

“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 “

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