How Can Algebraic Manipulation Prove the Consensus Property in Logic Circuits?

[shamieh]

Some background: I am in EE 280 Design of Logic Circuits.

Problem: Use algebraic manipulation to prove that xy +yz +x!z = xy + x!z. (Note that this is the consensus property which is: xy + yz + x!z = xy + x!z)

+ mean OR, ! mean NOT.

Please help! I am lost. I do have the rules near me (x AND 1 = x etc.. as well as the "Single Variable Theorems") If someone could walk me through solving this that would be great and I would be forever thankful.

 

[Ackbach]

Try this:
$$xy+\bar{x}z+yz=xy+\bar{x}z+(x+\bar{x})yz.$$

 

[shamieh]

Awesome, just figured it out. I solved and everything. I have a question tho. Where exactly does the rule (x + x!) come in play? I guess my question essentially is; how do I know that I can randomly put in a (x + x!) in the 2nd term. What is the property or rule that tells me I can do that legally? Sorry if this seems like a dumb question I'm just trying to understand what I'm actually doing versus just solving the equation.

Thanks again Ackbach,
-Sham(Ninja)

 

[Ackbach]

Well, $x+\bar{x}=\text{T}$, and $Tz=z$. So you can always multiply anything by $T$ and not change the truth value.