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+1 vote
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Let P and Q be two propositions $\sim(p\leftrightarrow Q)$ is equivalent to  :

$(1)P\leftrightarrow \sim Q \\ (2)\sim P\leftrightarrow Q \\ (3)\sim P \leftrightarrow \sim Q \\ (4)Q\rightarrow P$
in Discrete Mathematics by Veteran (432k points)
recategorized by | 691 views
0
Options 1 and 2 both are correct.
0
yes both are correct
0
Option 1) and 2) both are correct .

Right?
0
Yes, both 1st and 2nd are correct.
0
Why not option 1) ?
+1

P↔Q =  P Exnor Q 

So ~ ( P Exnor Q) = ~  P Exnor Q =  P Exnor ~Q = P Exor Q

Yes both are true.

5 Answers

+2 votes
Best answer
P Q $\sim p$ $\sim Q$ $Q \rightarrow P$ $\sim(P \leftrightarrow Q)$ $P \leftrightarrow \sim Q$
$\sim P \leftrightarrow Q$
$\sim P \leftrightarrow \sim Q$
0 0 1 1 1 0 0 0 1
0 1 1 0 0 1 1 1 0
1 0 0 1 1 1 1 1 0
1 1 0 0 1 0 0 0 1

Both option(1)and (2) is the correct choice.

by Boss (41.3k points)
selected by
0 votes
The answer will be both 1)P↔∼Q & 2)∼P↔Q
by Boss (17.8k points)
0 votes

....

by Boss (15k points)
0 votes
Option 1 is correct

Option 2 is also correct
by (303 points)
0 votes

$\sim$(P $\rightarrow$ Q)

= $\sim$( (P $\rightarrow$ Q) ^ (Q $\rightarrow$ P) )

= $\sim$ ( ($\sim$P v Q) ^ ( $\sim$Q v P) )

= $\sim$($\sim$P v Q) v $\sim$($\sim$Q v P)

= (P ^ $\sim$Q) v (Q v $\sim$P) = P$\bigoplus$Q

Using distributive law. (a + bc) = (a + b).(b + c)

= ( (P ^ $\sim$Q) v Q) ^ ( (P ^ $\sim$Q) v $\sim$P)

= ( ( P v Q)^($\sim$Q v Q)) ^ ( ( $\sim$P v  P) ^ ($\sim$Q v $\sim$P) )

= ( ( P v Q) ^ ( $\sim$Q v $\sim$ P)

Now here we have two choices.

1. ($\sim$P $\rightarrow$ Q) ^ ( Q $\rightarrow$ $\sim$P) = $\sim$P $\leftrightarrow$ Q

2. ($\sim$Q $\rightarrow$ P) ^ (P $\rightarrow$ $\sim$ Q) = P $\leftrightarrow$ $\sim$Q.

So both 1 and 2 option is correct.

by Boss (16.5k points)

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