Simplify the expression: $((XY+X')\oplus (Y+YZ))'$

Question

Simplify the expression: $((XY+X')\oplus (Y+YZ))'$

I can go as far as: $$((XY+X')\oplus (Y+YZ))'$$ $$\text{I use the rule of absorption:}$$ $$((Y+X')\oplus Y)'$$ Now I see that if $Y=1$ than the result is one and if $Y=0$ the result is $X$, but according to Wolfram the result is $X+Y$.

Where did I go wrong?

Answer 1

We know that: $$ A \oplus B = \bar A B + A \bar B$$ and that $\overline {A\cdot B} = \bar A+ \bar B $ and $\overline {A+B} = \bar A \bar B $. Thus, $$((Y + \bar X) \oplus Y) = \overline {Y + \bar X} Y + (Y + \bar X) \bar Y = (X\bar Y Y) + (Y\bar Y + \bar X \bar Y) = 0 + 0 + \bar X \bar Y $$

Thus, $$\overline {\bar X \bar Y} = X + Y $$ Hope it helps.

Answer 2

You didn't go wrong!

Notice that in general $X \oplus (X + Y) \Leftrightarrow X'Y$

(since $X$ cannot be true, as that would make both sides true ... and hence $Y$ should be true)

So, where you left off:

$$((Y+X')\oplus Y)' \Leftrightarrow$$

$$(X'Y')' \Leftrightarrow$$

$$X + Y$$