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Show that if xy = 0, then $x\oplus y$ = x + y.
in Digital Logic by Boss (10.8k points) | 68 views

2 Answers

+1 vote
if $xy = 0,$ then $x\oplus y = x + y$

We know that ${\color{Red}{x \odot y = x\cdot y+\overline{x}\cdot \overline{y}} }$  and  ${\color{Magenta}{{x\oplus y}=x\overline{y}+\overline{x}y}}$

                        put $x\cdot y =0$

           Then we get $x \odot y = 0+\overline{x}\cdot \overline{y}$

                             $x \odot y = \overline{x}\cdot \overline{y}$

We know that $x \odot y = x\cdot y+\overline{x}\cdot \overline{y}=\overline{x\oplus y}={\overline{x\overline{y}+\overline{x}y}}$

        Then now  $\overline{x\oplus y}=\overline{x}\cdot \overline{y}$

we can apply complement on both side

              $\overline{\overline{x\oplus y}}=\overline{\overline{x}\cdot \overline{y}}$

           $x\oplus y = \overline{\ \overline{x}}+ \overline{ \overline{y}}$

           $x\oplus y=x+y$           ${\color{DarkGreen}{\left [ \overline{ \overline{A}}=A\right]} }$
by Veteran (59.1k points)
+1 vote

we can solve using truth table 

by Boss (36.5k points)
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