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!
Best Answer
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):
$$(x_1 + x_2')(x_1 + x_3)$$