TIFR CSE 2015 | Part A | Question: 5

Question

What is logically equivalent to "If Kareena and Parineeti go to the shopping mall then it is raining":

• A. If Kareena and Parineeti do not go to the shopping mall then it is not raining.
• B. If Kareena and Parineeti do not go to the shopping mall then it is raining.
• C. If it is raining then Kareena and Parineeti go to the shopping mall.
• D. If it is not raining then Kareena and Parineeti do not go to the shopping mall.
• E. None of the above.

Comments

p->q is equivalent to its contrapositive(~q->~p).

so answer is D.

---

Proposition and it's contrapositive is logically equivalent

Answer 1

Answer will be (D)

"If Kareena and Parineeti go to the shopping mall then it is raining"

Let "Kareena and Parineeti go to the shopping mall" be represented by $p$ and "it is raining" by $q$

Now, the statement says that $p→q$

1. a. If Kareena and Parineeti do not go to the shopping mall then it is not raining.
   i.e., $\neg p→ \neg q$
   Not matching with the given implication.
2. b.If Kareena and Parineeti do not go to the shopping mall then it is raining.
   i.e., $\neg p→ q$
   Not matching with the given implication.
3. If it is raining then Kareena and Parineeti go to the shopping mall.
   i.e., $q→p$
   Not matching with the given implication.
4. If it is not raining then Kareena and Parineeti do not go to the shopping mall.
   i.e., $\neg q → \neg p \equiv q \vee \neg p \equiv p→q$
   Matces with the given implication.

So, correct option is (D).

Comments

Can you please exlain the (d) part

I didnt understood what formula applied for expansion there.

---

a- If Kareena and Parineeti do not go to the shopping mall then it is not raining.
The inverse of given conditional statement. ( ~p→ ~q)

c-  If it is raining then Kareena and Parineeti go to the shopping mall.
The converse of given conditional statement.  (q→p)

d- If it is not raining then Kareena and Parineeti do not go to the shopping mall.
The Contra-positive of given conditional statement.  (~q → ~p )

Given conditional statement and its contrapositive are always logically equivalent.
 ~q → ~p = q v ~p = p→q

Inverse and converse of a conditional statement are always logically equivalent.
~p→~q = p v ~q = q → p

---

Let "Kareena and Parineeti go to the shopping mall" be represented by p is wrong according to me as kareena and parineeti are two different entities and not a single entity.

Let P: Kareena go to the shopping mall

Q: Parineeti go to the shopping mall

R: It is raining

Now, the statement says that p ∧ q→r
so contrapositive statement would be

r’ → (p ∧ q)’

=> r’  → p’ + q’

=> If it is not raining then either Kareena did not go to the shopping mall or Parineeti did not go to the shopping mall.

therefore, Answer: E – None of the above.

 

---

Kareena and Parineeti -- both should go for shopping as per the given statement. So they are indeed considered as a single entity.

If Sachin and Sehwag opens, India wins.

Now, this is equivalent to

If India loses, Sachin and Sehwag didn't open.

If only one of them opens, India may win or lose.

Answer 2

p->q is equivalent with its contrapositive (Hence Option D is Ans.)

Note:-Contrapositive of p->q means converse then inverse of p->q

or inverse then converse of p->q

Answer 3

A different take on the question:

K: Kareena goes to the mall

P: Parineeti goes to the shopping mall

R: It is raining

Premise: $K \wedge P \rightarrow R$

The contrapositive of the premise is: ⌉R -> ⌉K V ⌉P

If it is not raining then Kareena does not go to the mall or Parineeti does not go to the mall.

Answer 4

Let P: Kareena go to the shopping mall

Q: Parineeti go to the shopping mall

R: It is raining

 

Now, the statement says that p ∧ q→r

so contrapositive statement would be

r’ → (p ∧ q)’

=> r’  → p’ + q’

=> If it is not raining then either Kareena did not go to the shopping mall or Parineeti did not go to the shopping mall.

therefore, Answer: E – None of the above.