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

Your application of what you call "commutativity" is incorrect. Variables commute only over the same operations.
Instead: $$\overline{a} \cdot \overline{b} \cdot \overline{c} +a\cdot b+ a \cdot \overline{c}= \overline{a} \cdot \overline{b} \cdot \overline{c} + a\cdot (b+\overline c) $$
$$= \overline{a} \cdot \overline{b} \cdot \overline{c} + b + \overline c$$
$$ = (\overline a \cdot \overline b + 1)\cdot \overline c + b$$
$$=  1\cdot \overline c + b$$
$$ = \overline c + b$$
