# Algebraic Manipulation

• Sep 12th 2013, 07:48 AM
shamieh
Algebraic Manipulation
This is Boolean Algebra. + means OR, the . means AND and the x! means X NOT.

Use algebraic manipulation to prove that x +yz = (x + y).(x+z)

x + yz = (x+y) AND (x + z) ...

x +yz = xx + xz + xy + xy + yz
= x(x + z + y) + yz
= x + z + y = 1
= x AND 1 = x
x +yz =x + yz

is that correct?

Next problem: prove that (x+y)(x+y!) = x
= xx + xy! + xy + yy!
= x(x + y! + y) + yy!
what does x + y! + y evaluate to? is it 1?
in that case;
=x AND 1 = x
But what does yy! evaluate to?

I also need to use algebraic manipulation to prove that xy + yz +x!z = xy + x!z using the consensus property.

Sham
• Sep 12th 2013, 09:09 AM
johng
Re: Algebraic Manipulation
Hi Sham,
Proving Boolean algebra equations is just like proving other equations. You should start with one side, and, by a chain of equations, reach the other side.

(x+y)(x+z) = xx+xz+xy+yz = x + xz +xy +yz = x(1 + z +y) + yz = x1+yz = x + yz

(x+y)(x+y!) = xx +xy +xy! +yy! = x + xy +xy! + 0 = x(1+y+y!) = x1 = x
• Sep 12th 2013, 09:32 AM
shamieh
Re: Algebraic Manipulation
John, thanks so much! But what is (1 +y + y!) = to? I know y*y! = 0 but what about (1+y+y!)?
• Sep 12th 2013, 09:50 AM
Shakarri
Re: Algebraic Manipulation
y OR y! is 1. If y is false then not y is true

Also 1 OR [anything]=1
Do you understand the OR opperation?
X OR Y is asking if either X is true or Y is true. Therefore 1 OR X is asking if either 1 is true or X is true. 1 is always true so regardless of the value of X: 1 OR X is always true
• Sep 12th 2013, 09:57 AM
johng
Re: Algebraic Manipulation
Hi Shamieh,
In Boolean algebra 1 + anything = 1; i.e. true or anything is true no matter what the truth value of anything.
• Sep 12th 2013, 07:15 PM
shamieh
Re: Algebraic Manipulation
Quote:

Hi Shamieh,
In Boolean algebra 1 + anything = 1; i.e. true or anything is true no matter what the truth value of anything.
So for example, x(1 + x +y +x +y +x +z +x +y) you are saying that that will evaluate to 1? Getting x AND 1 which evaluates to x for example? And I'm not sure what shakari is talking about because y AND y! definitely evaluates to 0. Unless my book is lying...