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Consider the following boolean equations:

  1. $wx+w(x+y)+x(x+y)=x+wy$
  2. $(w \overline{x}(y+x \overline{z})+ \overline{w} \overline{x})y= \overline{x}y$

What can you say about the above equations ?

  1. (i) is true and (ii) is false
  2. (i) is false and (ii) is true
  3. Both (i) and (ii) are true
  4. Both (i) and (ii) are false
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1 Answer

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$1\,)\,wx+w(x+y)+x(x+y)$

      $=wx+wx+wy+x+xy$

      $=wx+wy+x(1+y)=wx+wy+x$

      $=x(1+w)+wy=x+wy$

So, $1$ is true.

$2\,)\,(w\bar{x}(y+x\bar{z})+\bar{w}\,\bar{x})y$

       $=(w\bar{x}y+wx\bar{x}\bar{z}+\bar{w}\bar{x})y$

       $=(\bar{x}(\bar{w}+wy)+0)y\,\,\,\left \{A\,\bar{A}=0\right \}$

       $=(\bar{x}(\bar{w}+y))y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left \{A+\bar{A}B=A+B\right \}$

       $=\bar{x}\bar{w}y+\bar{x}y=\bar{x}y(\bar{w}+1)$

       $=\bar{x}y$

So, $2$ is true

$C$ should be the answer

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