# Simplify the boolean expression

larry21

## Homework Statement

Build the truth tables for the boolean expressions.

(x + y' * z)' + (x * y * z' + (y + z)') + (y + (x * z')')'

## The Attempt at a Solution

Given that the expression is so complex I believe that I need to simplify first then proceed to build the truth table?

Work so far:
(x + y' * z)' + (x * y * z' + (y + z)') + (y + (x * z')')'

1.) x' * y'' + z' + (x * y * z' + y' + z') + y' * (x * z')''
2.) x' * y + z' + (x * y * z' + y' + z') + y' * x * z'

Steps:
1. De Morgan
2. De Morgan and double negation

Kind of stuck at this point....
Help would be appreciated. Thanks in advance.

## Answers and Replies

Homework Helper
Gold Member
If it were me, I'd skip the simplification and go straight to the truth table. There's only three input variables (x, y and z), so your truth table has only eight rows.

There's nothing stopping you from simplifying beforehand, but as it turns out, going straight to the truth table is often the easiest approach.

The reason I say this is because the usual approach to minimization is:
(1) Build the truth table
(2) Use the truth table to make a Karnaugh map (K-map) and simplify that way (or use the Quine–McCluskey algorithm or equivalent).

Simplifying before the truth table sort of defeats the point. Sure, you can do it, but it might not be the best way to expend your effort.

Last edited:
• 1 person
Homework Helper
Gold Member
That said,

If you do choose to simplify a little before making the truth table, I suggest trying to put the expression in the form of "Sum of Products." It makes it a little easier to fill in the table that way. It's not totally necessary, but it couldn't hurt.

There are a couple of mistakes below I think.

Work so far:
(x + y' * z)' + (x * y * z' + (y + z)') + (y + (x * z')')'

1.) x' * y'' + z' + (x * y * z' + y' + z') + y' * (x * z')''
2.) x' * y + z' + (x * y * z' + y' + z') + y' * x * z'
I think you mean, (changes in red)

1.) x' * (y'' + z') + (x * y * z' + y' * z') + y' * (x * z')''
2.) x' * (y + z') + (x * y * z' + y' * z') + y' * x * z'