Simplifying Boolean Logic Relation

Question

Simplification of $X \ \ \overline{(\overline{X} Y + \overline{Y}Z)} +\overline{X}$

The answer found is $\overline{X}+Y+\overline{Z}$

I tried the following approach, but ends up in two term expression.

$X(X\overline{Y}Y\overline{Z})+\overline{X}(Y+\overline{Y})(Z+\overline{Z})$

$X\overline{Z}+\overline{X}YZ+\overline{X} \overline{Y}Z +\overline{X}Y\overline{Z}+\overline{X} \overline{Y}\overline{Z}$

$X\overline{Z}+\overline{X}Y+\overline{X} \overline{Y}$

$X\overline{Z}+\overline{X}$

The answer should be $\overline{X}+Y+ \overline{Z}$. What was the error and how to derive the final answet. Any clue please?

Answer 1

You used the De-Morgan's laws wrong.
Recall: $$(I) ~~~ \overline{X+Y} = \overline X \cdot \overline Y \\(II) ~~~ \overline{X \cdot Y} = \overline X + \overline Y$$

And so we begin to solve, using De-Morgan's law (I):

$$X \cdot ~ \overline{( \overline {X} \cdot Y + \overline{ Y} \cdot Z)} + \overline X = \\ X \cdot ( \overline{(\overline X Y)} \cdot \overline{( \overline Y Z))} + \overline X$$

This is the point where you used it wrong, now we use it again:

$$X \cdot ( \overline{(\overline X Y)} \cdot \overline{( \overline Y Z))} + \overline X \\ = X \cdot (X + \overline Y) \cdot (Y + \overline Z) + \overline X$$

Using the absorption law:

$$ A( A+ B) = A$$

We can simplify this to:

$$ X \cdot(Y + \overline Z) + \overline X$$

And now finishing with the known fact (which isn't that hard to prove too, look here: In boolean algebra, why is a+a'b = a+b?):

$$ A + \overline A B = A+B$$

And substituting $ A = \overline X , ~~ B = Y + \overline Z$ to get:

$$ \overline X + Y + \overline Z$$

Answer 2

$X \ \ \overline{(\overline{X} Y + \overline{Y}Z)} +\overline{X} = X \ \ ({\overline{\overline{X} Y} \, \, \overline{\overline{Y}Z)}} +\overline{X}$

$ = X \ \ ({({X + \overline{Y}}) \, \, (Y + \overline{Z)}}) +\overline{X}$

$ = X \ \ ({XY + X \overline{Z}} + \overline{Y} \, \overline{Z}) +\overline{X}$

$ = XY + X \overline{Z} + X \overline{Y} \, \overline{Z} + \overline{X}$

$ = \overline{X} + Y + \overline{Z}$

Pls note $XY + \overline{X} = \overline{X} + Y$, $X \overline{Z} + X \overline{Z} \, \overline{Y} = X \overline{Z}$ and $\overline {X} + X \overline {Z} = \overline {X} + \overline {Z}$