Boolean Algebra: Is this equality or inequality?

Question

Consider: $$xy + x'y' + yz = xy + x'y' +x'z$$

Is this equality true?

I know I could a truth-table but I prefer doing it algebraically. I think there's something tricky here (Like adding a term, relying on the fact that $x+x = x$).

Answer 1

No real tricks, but it does use $X + X = X$ (Idempotent) and $X + \bar X = 1$ (Complement). $$xy + \bar x \bar y + yz = xy (\bar z + z) + \bar x \bar y(\bar z + z) + yz (\bar x + x)$$ Expand missing terms. $$= xy\bar z + {\color{Red} {xyz}} + \bar x\bar y\bar z + \bar x\bar yz + \bar xyz + {\color{Red} {xyz}}$$ Remove duplicates. $$= xy\bar z + xyz + \bar x\bar y\bar z + \bar x\bar yz + \bar xyz $$ Look for common terms - duplicate as needed - looking for $\bar xz$.

$$= (xy\bar z + xyz) + (\bar x\bar y\bar z + {\color{Red} {\bar x\bar yz}}) + (\bar xyz + {\color{Red} {\bar x \bar yz}}) \\ $$ $$= xy (\bar z + z) + \bar x\bar y (\bar z + z) + \bar xz (y + \bar y) $$

So they are equal. $$xy + \bar x\bar y + yz = xy + \bar x\bar y + \bar xz$$