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

  1. If Kareena and Parineeti do not go to the shopping mall then it is not raining.
  2. If Kareena and Parineeti do not go to the shopping mall then it is raining.
  3. If it is raining then Kareena and Parineeti go to the shopping mall.
  4. If it is not raining then Kareena and Parineeti do not go to the shopping mall.
  5. None of the above.
in Mathematical Logic by Boss (30.8k points)
edited by | 521 views
0
p->q is equivalent to its contrapositive(~q->~p).

so answer is D.
0
Proposition and it's contrapositive is logically equivalent

3 Answers

+27 votes
Best answer

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).

by Veteran (119k points)
edited by
0
Can you please exlain the (d) part

I didnt understood what formula applied for expansion there.
+1

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

+7 votes

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

by Boss (23.9k points)
+2 votes

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.

by Junior (879 points)

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