Simplification of expression $\bar{x}\bar{y}\bar{z} + \bar{x}\bar{y}z+x\bar{y}\bar{z}+x\bar{y}z$

Question

Here are my steps: $\bar{x}\bar{y}\bar{z} + \bar{x}\bar{y}z+x\bar{y}\bar{z}+x\bar{y}z$ $$\bar{x}\bar{y}\bar{z} + \bar{x}\bar{y}z+x\bar{y}\bar{z}+(x\bar{y}z+x\bar{y}z)$$ Used Idempotent law and rearranged the equation $$\bar{x}\bar{y}\bar{z} + (\bar{x}\bar{y}z+x\bar{y}z)+(x\bar{y}\bar{z}+x\bar{y}z)$$ factoring $$\bar{x}\bar{y}\bar{z} + \bar{y}z(\bar{x}+x)+x\bar{y}(\bar{z}+z)$$Complement law $$\bar{x}\bar{y}\bar{z} + \bar{y}z(1)+x\bar{y}(1) + \bar{x}\bar{y}\bar{z} $$ Used Idempotont law again $$\bar{x}\bar{y}\bar{z} + \bar{y}(x + \bar{x}+z+\bar{z}) $$factoring $$\bar{x}\bar{y}\bar{z} + \bar{y}(1+1) $$Complement Law and Tautology $$\bar{x}\bar{y}\bar{z} + \bar{y}1$$Used Identity law on y $$\bar{y}(1+\bar{x}\bar{z})$$Tautology $$\bar{y}$$ However I need x and z values. $$\bar{y}+\bar{x}x+z\bar{z}$$ The question states that I need to find the minimization of the original expression, as the sum of three terms. Is my steps valid?

Answer 1

1. Your simplification process is correct, and so is your result $\bar y$.
2. If your truth tables don't match then you probably made a mistake computing the truth tables. I can't check those because you haven't posted them.
3. Your last statement "The question states that I need to find the minimization of the original expression, as the sum of three terms." is very strange since the expression simplifies to just 1 term. I realize you can awkwardly rewrite it as 3, but I would expect whoever assigned this question would not ask for 3 terms if the expression simplifies to just $\bar y$. It makes me think you should double-check for mistakes in copying down the initial expression to be simplified. (That could also explain why your truth tables aren't matching up.)