UGC NET CSE | November 2017 | Part 2 | Question: 8

Question

Let $P$ and $Q$ be two propositions $\neg (P \leftrightarrow Q)$ is equivalent to

• A. $P\leftrightarrow \neg Q$
• B. $\neg P\leftrightarrow Q$
• C. $\neg P \leftrightarrow \neg Q$
• D. $Q\rightarrow P$

Comments

3..?

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can u explain how option 3 @  MiNiPanda

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p<->q means (p->q)•(q->p)

= (!p+q)(!q+p) = p xnor q.

From the options given you can check that option 1 and 2 will give (p xor q) and last one will give (!q+p). Only option 3 is giving pxnorq.

Answer 1

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.

Answer 2

$¬(P\leftrightarrow Q)$

$ ¬(P\rightarrow Q \wedge Q\rightarrow P)$

$¬ ((\bar{P}\vee Q) \wedge (\bar{Q}\vee P))$

$¬(\bar{P}\bar{Q} \vee PQ)$

$¬(P \odot Q)$

$(P\oplus Q)$

now check which options looks like XOR 

P	$\bar{P}$	Q	$\bar{Q}$	$\bar{P} \leftrightarrow Q$	$P \leftrightarrow \bar{Q}$
T	F	T	F	F	F
T	F	F	T	T	T
F	T	T	F	T	T
F	T	F	T	F	F

$\bar{P} \leftrightarrow Q$ and $P \leftrightarrow \bar{Q}$ both are correct

Answer 3

The answer will be both 1)P↔∼Q & 2)∼P↔Q

Answer 4

....

Answer 5

Option 1 is correct

Option 2 is also correct

Answer 6

$\sim$(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.