Boolean Simplification of $(\overline{a+b+c})+a\cdot(b+ \overline{c})$

Question

I'm lost, when checking my answer via truth tables, my simplified form does not match the original equation. My work, with reasoning step by step is below. Can you help me figure out where I'm wrong, and what it should be?

$(\overline{a+b+c})+a\cdot(b+ \overline{c})$ $\quad $ original equation

$(\overline{a+b+c})+a\cdot b+ a \cdot \overline{c}$ $\quad$ Distribution

$(\overline{a} \cdot \overline{b} \cdot \overline{c}) +a\cdot b+ a \cdot \overline{c}$ $\quad$ DeMorgan

$(\overline{a} \cdot \overline{b}) \cdot \overline{c} +a\cdot b+ a \cdot \overline{c}$ $\quad$ Associativity

$(\overline{a} \cdot \overline{b}) +a\cdot b \cdot \overline{c} + a \cdot \overline{c}$ $\quad$ Commutativity

$\overline{c} + a \cdot \overline{c}$ $\quad $ Complement

$\overline{c}$ $\quad $ Absorption

Answer 1

Your application of what you call "commutativity" is incorrect. Variables commute only over the same operations.

Instead: $$\overline{a} \cdot \overline{b} \cdot \overline{c} +a\cdot b+ a \cdot \overline{c}= \overline{a} \cdot \overline{b} \cdot \overline{c} + a\cdot (b+\overline c) $$

$$= \overline{a} \cdot \overline{b} \cdot \overline{c} + b + \overline c$$ $$ = (\overline a \cdot \overline b + 1)\cdot \overline c + b$$

$$= 1\cdot \overline c + b$$

$$ = \overline c + b$$

Answer 2

If you have one valuation for $a$, $b$ and $c$ where the initial expression and your final result disagree, see what your intermediate expressions evaluate to for that particular valuation. When the value changes you have found at least one place where your rewriting is not valid.

(More concretely, the rewriting you justify as "commutativity" looks fishy. You would need something like $xy+z=x+zy$ for that, but that doesn't have anything to do with commutativity. And I don't understand at all what's happening at "complement".)