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+19 votes

What is the converse of the following assertion?

  • I stay only if you go
  1. I stay if you go
  2. If I stay then you go
  3. If you do not go then I do not stay
  4. If I do not stay then you go
asked in Mathematical Logic by Veteran (52.1k points)
edited by | 2.9k views

P  = "I stay" , Q = "you go"
question is P--->Q then option A is
i stay(P) <---- you go(Q)  , which means Q implies P

also Q---->P 


I stay only if you go

    P: I stay

   Q: You go

It is given in the form of P if Q which is equivalent to (P→ Q), we can write that statement in this way,

If I stay then you go.

Along with that, we know that converse of (P→Q) is (Q→P)

we can write that converse in the English language is If you go then I stay and we can write this (Q→P) form to Q if P form.

which means If you go then I stay is equivalent to I stay if you go.

So the answer is option A

what does converse mean here?

Conditional statement$: P\rightarrow Q\equiv\neg P\vee Q,$ means If $P$ then $Q$ $(or)$  $P$ only if $Q$ $(or)$ $Q$ if $P$

Converse$:Q\rightarrow P\equiv\neg Q\vee P$

Inverse$:\neg P\rightarrow\neg Q\equiv P\vee\neg Q$

Contrapositive$:\neg Q\rightarrow \neg P\equiv Q\vee\neg P\equiv \neg P\vee Q$


$P\rightarrow Q$  can also be written as 

$\bullet$  if P then Q

$\bullet$  If P, Q

$\bullet$  Q if P

$\bullet$  Q when P

$\bullet$  Q unless ~P

$\bullet$  P only if Q

$\bullet$  P implies Q

$\bullet$  Q whenever P

$\bullet$  Q follows from P


This is also possible for $P\rightarrow Q$

$P$ is sufficient for $Q$

A necessary condition for $P$ is $Q$

$Q$ is necessary for $P$

A sufficient condition for Q is $P$


3 Answers

+25 votes
Best answer

"I stay only if you go" is equivalent to "If I stay then you go."

 $\because A \text{ only if } B  \equiv  (A \to B) $

$A=$ "I stay" and $B=$ "You go"

Converse $( A\to B)  =  B\to A$

"If you go then I stay"

Answer is  (A).

answered by Active (2.6k points)
edited by
Shouldn't it be B<->A because of "only if" statement. It is same as "if and only if".

I think answer is B. 

I stay only if you go 

A = You go

B= I stay

So, given is A->B

Converse is B->A

So, ans is 

If I stay, then you go.

So, ans is B. 

Please explain me if I am wrong.


I stay only if you go  is equivalent to if I stay then you go. check again

Now I understood the meaning of "only if".... I thought A only if B means B->A.

But, A only if means A->B,

So ans is A.

Thanks for the explanation Anirudh...

A only if B $\equiv$ A $\rightarrow$ B

Given : A $\rightarrow$ B  . We have to find all which given below for A $\rightarrow$ B ?

Direct = A $\rightarrow$ B

Converse = B $\rightarrow$ A

Inverse =  ~A $\rightarrow$ ~B

COntrapositive = ~B $\rightarrow$ ~A

0 votes
option a.  converse is B->A can also be written as "A if B" thats u ans (a)
answered by Junior (747 points)
0 votes

I stay only if you go is equivalents to   If I stay then you go.

  A only if B  => A->B

A= "I stay" and B= "You go"
i.e.Conditional: A ---.>B 
     Converse:   B----->A

" If you go then I stay "

Answer is  (A) A if B is equivalence to B -------->A
         "I stay if you go" is also equivalent to "If you go then I stay"

i.e.Conditional: P ---.>Q 
     Converse:  Q----->P 

answered by Boss (42.3k points)
edited by

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