Boolean Algebra Manipulation/Simplification I have come across a couple questions while doing my digital logic work.
1) Is it possible to simplify these, while keeping each a product of sums? (I'm leaning towards no--the only way I could see to simplify them would be to distribute.)  They're separate problems.
$$(a+b+c)(a'+b'+c')$$
$$(x+y)(x'+y+z')$$
2) Find the minimum sum of products expression (I honestly didn't even know how to begin this one, if you could just get me started...):
$$x_1'x_3'x_5'+x_1'x_3'x_4'+x_1'x_4x_5+x_1x_2'x_3'x_5$$
- The hint was to use the consensus theorem: $xy+yz+x'z=xy+x'z$
3) Find the minimum product of sums expression (again, if you could just help me get started)
$$x_1x_3'+x_1x_2+x_1'x_2'+x_2'x_3$$
Any help is greatly appreciated!  Thanks!
 A: 1b)
$$(x+y)(x'+y+z')$$
can be simplified to
$$(x+y)(x'+z')$$
Convince yourself using a thruth table.
2)
$$x_1'x_3'x_5'+x_1'x_3'x_4'+x_1'x_4x_5+x_1x_2'x_3'x_5$$
can be simplified to
$$x_1'x_3' + x_1'x_4x_5 +  x_2'x_3'x_5$$
Such simplifications can be done using a Karnaugh-Veitch map.
3)
$$x_1x_3'+x_1x_2+x_1'x_2'+x_2'x_3$$
is a sum of products. 
It can be minimized to
$$x_1 + x_2'x_3$$
Written in Conjunctive Normal Form (CNF):
$$(x_1 + x_2')(x_1 + x_3)$$
A: 1b) 
$$(x+y)(x'+y+z')=\text{ (Distribution)}$$
$$y+x(x'+z')= \text{ (Reduction)}$$
$$y +xz'$$
This is actually an instance of a General Reduction Theorem that says that:
General Consensus Theorem
$PQ+PQ'R=PQ+PR$
$(P +Q)(P+Q'+R)=(P+Q)(P+R)$
We can also use this for 2):
$$x_1'x_3'x_5'+x_1'x_3'x_4'+x_1'x_4x_5+x_1x_2'x_3'x_5=\text{ (Distribution)}$$
$$x_1'(x_3'x_5'+x_3'x_4'+x_4x_5)+x_1x_2'x_3'x_5=\text{ (Consensus)}$$
$$x_1'(x_3'x_5'+x_3'x_4'+x_4x_5+x_3'x_5)+x_1x_2'x_3'x_5=\text{ (Adjacency)}$$
$$x_1'(x_3'+x_3'x_4'+x_4x_5)+x_1x_2'x_3'x_5=\text{ (Absorption)}$$
$$x_1'(x_3'+x_4x_5)+x_1x_2'x_3'x_5=\text{ (Distribution)}$$
$$x_1'x_3'+x_1'x_4x_5+x_1x_2'x_3'x_5=\text{ (General Reduction)}$$
$$x_1'x_3'+x_1'x_4x_5+x_2'x_3'x_5$$
Finally, the Consensus Theorem also works great for 3):
$$x_1x_3'+x_1x_2+x_1'x_2'+x_2'x_3=\text{ (Consensus)}$$
$$x_1x_3'+x_1x_2+x_1'x_2'+x_2'x_3+x_1x_2'=\text{ (Adjacency)}$$
$$x_1x_3'+x_1+x_2'+x_2'x_3=\text{ (Absorption)}$$
$$x_1+x_2'$$
