Boolean Expression Simplification

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In summary: So basically just keep trying until it works then?In summary, the Laws of Boolean Algebra were used to simplify the expressions (x'y'+xy'+x'y) and (p+q'p)(p+qr). The first expression was simplified to x'+y' and the second expression was simplified to p. The process involved noticing and applying various rules and tricks to manipulate the expressions.
  • #1
mad_monkey_j
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Homework Statement


Simplify the following expressions
(x'y'+xy'+x'y)

(p+q'p)(p+qr)


Homework Equations


Laws of boolean algebra

The Attempt at a Solution


For the first one I've found many ways to solve it... something isn't right here
(x'y'+xy'+x'y)
=x'(y'+y)+xy'
=x'+xy'
=x'+x+y'
=y'

OR

=y'(x'+x)+x'y
=y'+x'y
=y'+x'+y
=x'
I'm definantly doing something wrong here... probably making up laws :biggrin:

For the second I'm not sure if the last line is even possible
(p+q'p)(p+qr)
=pp+pqr+ppq'+q'qr
=p+pqr+pq'+0
=p+pqr
=p? Can I do that with pqr having 3 variables?
 
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  • #2
mad_monkey_j said:
=x'+xy'
=x'+x+y'
=y'
That is wrong. Apply:

a+a'b=a+b

For the second question, replace the fist factor by p+q'p=p. You made a mistake in the derivation, but the result is p.

ehild
 
  • #3
Ahh so the first is
=x'+y'

and with the second if I change it to:
p(p+qr)
won't I end up with
=p+pqr in the end anyway?

Also how do you know what rule you're supose to use? Is there a specific order or is it just whatever you notice first?
 
  • #4
mad_monkey_j said:
and with the second if I change it to:
p(p+qr)
won't I end up with
=p+pqr in the end anyway?

I said the result was correct. You made a mistake as the last term was q'qpr, but it canceled because of q'q.

mad_monkey_j said:
Also how do you know what rule you're suppose to use? Is there a specific order or is it just whatever you notice first?

There are no general recipes.
Use what you notice first, but it helps if you notice when something cancels, for example E+a=E, a+a'=E, aa'=0. Factor out what you can, so a+ab=a(E+b)=a. But it is not a rule; sometimes you need to apply other tricks.

ehild
 
  • #5
ehild said:
I said the result was correct. You made a mistake as the last term was q'qpr, but it canceled because of q'q.



There are no general recipes.
Use what you notice first, but it helps if you notice when something cancels, for example E+a=E, a+a'=E, aa'=0. Factor out what you can, so a+ab=a(E+b)=a. But it is not a rule; sometimes you need to apply other tricks.

ehild

Ahh fair enough, thanks for the help.
 

1. What is a Boolean expression?

A Boolean expression is a mathematical expression that consists of variables, logical operators, and parentheses, and evaluates to either true or false. It is commonly used in computer programming and digital logic to make decisions based on logical conditions.

2. What is Boolean expression simplification?

Boolean expression simplification is the process of reducing a complex Boolean expression to its simplest form. This involves applying mathematical laws and rules to remove unnecessary variables, operators, and parentheses, resulting in a more concise expression that still evaluates to the same truth value.

3. Why is Boolean expression simplification important?

Boolean expression simplification is important because it can make complex logic easier to understand and more efficient to implement. It also helps to reduce the chances of errors and makes debugging and troubleshooting easier.

4. What are the steps involved in Boolean expression simplification?

The steps involved in Boolean expression simplification are as follows:

  • 1. Identify the terms and operators in the expression.
  • 2. Use Boolean algebra laws and rules to simplify the expression.
  • 3. Combine like terms and eliminate redundant terms.
  • 4. Use De Morgan's laws to simplify expressions with negations.
  • 5. Repeat until the expression is in its simplest form.

5. What are some common mistakes to avoid when simplifying Boolean expressions?

Some common mistakes to avoid when simplifying Boolean expressions include:

  • 1. Forgetting to distribute negations using De Morgan's laws.
  • 2. Incorrectly applying Boolean algebra laws and rules.
  • 3. Neglecting to combine like terms.
  • 4. Forgetting to eliminate redundant terms.
  • 5. Failing to check the final expression for correctness.

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