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Let $\oplus$ denote the exclusive OR (XOR) operation. Let '1' and '0' denote the binary constants. Consider the following Boolean expression for $F$ over two variables $P$ and $Q$:

$$F(P,Q)=\left( \left(1 \oplus P \right) \oplus \left( P \oplus Q \right )\right ) \oplus \left(\left(P \oplus Q\right) \oplus \left(Q \oplus 0\right)\right)$$

The equivalent expression for $F$ is

  1. $P+Q$
  2. $\overline{P+Q}$
  3. $P \oplus Q$
  4. $\overline {P \oplus Q}$
asked in Digital Logic by Veteran (101k points) | 1.8k views
+3

As ⊕ is commutative and associative operator, so we can remove all the parentheses if required, and apply the following three properties
1. x⊕x = 0
2. 0⊕x = x
3. 1⊕x= x'

Answer will be (D)

4 Answers

+24 votes
Best answer
XOR is associative and commutative. Also, $A \oplus A = 0$ and $A \oplus 1 = \overline{ A}$ and $A \oplus 0 = A$.  So
$\left( \left(1 \oplus P \right) \oplus \left( P \oplus Q \right )\right ) \oplus \left(\left(P \oplus Q\right) \oplus \left(Q \oplus 0\right)\right)$
$\implies \left(1 \oplus P \right) \oplus \left( \left( P \oplus Q \right ) \oplus \left(P \oplus Q \right) \right) \oplus \left(Q \oplus 0\right)$
$\implies  \left(1 \oplus 0 \right) \oplus \left( P \oplus Q \right) $
$\implies 1 \oplus \left(  P\oplus Q \right)$
$\implies \overline {\left( P \oplus Q\right)}$
answered by Veteran (357k points)
edited
0

@Arjun pls explain this simplification 

0

⟹(1⊕P)⊕((P⊕Q)⊕(P⊕Q))⊕(Q⊕0)    //Sir is the expression in bold = 0 ? can u elaborate a bit ?
⟹(1⊕0)⊕(P⊕Q)

0

@Arjun Sir, in your second step: 

⟹(1⊕P)⊕((P⊕Q)⊕(P⊕Q))⊕(Q⊕0)⟹(1⊕P)⊕((P⊕Q)⊕(P⊕Q))⊕(Q⊕0)
⟹(1⊕0)⊕(P⊕Q)

=(P') ⊕ (0) ⊕ (Q)

=((P') ⊕ (0) )⊕ (Q)

=P' ⊕ Q

=P ExNOR Q

=(P⊕Q)'

0
@Meghna yes, it is correct.

@Syed Yes, $A \oplus A = 0,$ here $A = (P \oplus Q)$
+9 votes

D)
Since there are only 2 variables putting in pair of values of P and Q in F and checking with the options is a time saving method.
But Lets solve it.


 

answered by Active (3.6k points)
0 votes
observe the common term p ex or q in both

consider a case where p and q are equal

then p ex or q results in 0

1)in first p is ex ored  with 1

2)in second q is ex ored with 0

so if p, q are same then either of one oresults in 1 and another to 0

1 ex or  0

it is exnor
answered by (395 points)
0 votes

We need to simplify the above expression. As the given operation is XOR, we shall see property of XOR. Let A and B be boolean variable. In A XOR B, the result is 1 if both the bits/inputs are different, else 0. Now,

( ( 1 X P) X (P X Q) ) X ( (P X Q) X (Q X 0) )

( P' X P X Q ) X ( P X Q X Q ) ( as 1 X P = P' and Q X 0 = Q )

(1 X Q) X ( P X 0) ( as P' X P = 1 , and Q X Q = 0 )

Q' X P ( as 1 X Q = Q' and P X 0 = P )

PQ + P'Q' ( XOR Expansion, A X B = AB' + A'B )

This is the final simplified expression.

Now we need to check for the options.

If we simplify option D expression.

( P X Q )' = ( PQ' + P'Q )' ( XOR Expansion, A X B = AB' + A'B )

((PQ')'.(P'Q)') ( De Morgan's law )

( P'+ Q).(P + Q') ( De Morgan's law )

P'P + PQ + P'Q' + QQ'

PQ + P'Q' ( as PP' = 0 and QQ' = 0 ) 

Hence both the equations are same. Therefore Option D. 
answered by Loyal (8.4k points)
0

you know the property from truth table of xor gate that is A(xor)0=A and A(xor)1=A'  A(xor)A=0 apply this rule on the above expression  

F(P,Q)=(1XOR P)XOR(PXORQ)XOR(PXORQXORQXOR 0)

=P'XOR(PXORQ)XOR(PXORQXORQ)

=P'XORPXORQXORP

=1XORQXORP

=Q'XORP

=(Q+P)'1



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