# 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!

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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):

$$(x_1 + x_2')(x_1 + x_3)$$

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