# Thread: Prove using mathematical proof

1. ## Prove using mathematical proof

Prove using mathematical proof:

$xyz + \overline{x}z = yz + \overline{x}z$

I can't figure how to begin with this. Help appreciated. thanks.

2. ## Re: Prove using mathematical proof

Originally Posted by larry21
Prove using mathematical proof:

$xyz + \overline{x}z = yz + \overline{x}z$

I can't figure how to begin with this. Help appreciated. thanks.
Prove using boolean postulates, laws, theorems. Can anyone help me? thanks.

3. ## Re: Prove using mathematical proof

$xy+\bar{x} = xy+\bar{x}(y+1) = xy+\bar{x}y+\bar{x} = (x+\bar{x})y+\bar{x} = y+\bar{x}$.

4. ## Re: Prove using mathematical proof

Originally Posted by emakarov
$xy+\bar{x} = xy+\bar{x}(y+1) = xy+\bar{x}y+\bar{x} = (x+\bar{x})y+\bar{x} = y+\bar{x}$.
Sorry, I don't quite follow what you just did. I need to prove that the left side: $xyz + \overline{x}z$ is equal to the right side: $yz + \overline{x}z$

5. ## Re: Prove using mathematical proof

Originally Posted by emakarov
$xy+\bar{x} = xy+\bar{x}(y+1) = xy+\bar{x}y+\bar{x} = (x+\bar{x})y+\bar{x} = y+\bar{x}$.
Originally Posted by larry21
Sorry, I don't quite follow what you just did. I need to prove that the left side: $xyz + \overline{x}z$ is equal to the right side: $yz + \overline{x}z$
Multiply both the LHS and RHS by z.

-Dan

6. ## Re: Prove using mathematical proof

Very sorry for being unclear in my question and causing such confusion. But I only require to prove the LHS of the equation, thus by using the boolean postulates, laws, theorems to collapse the LHS to the RHS. I've studying this equation for so long and have yet been able to achieve such a task. A complete solution is not required, I just need a starting point to guide me in the right direction. Thank you in advance for your help.

Edit:
$xyz + \overline{x}z = yz + \overline{x}z$

z(xy + x') = distributive law (factor)
z(x' + x)(x' + y) = distributive law (expand)
z(1)(x' + y) = complement
z(x'+y) = (anything ANDed by 1 equals itself)
zx' + zy = distributive (expand)
yz + x'z = commutative

I believe I've finally figured it out. Anyone care to critique? Thank you.

7. ## Re: Prove using mathematical proof

As you can note, the key part of your solution is converting xy + x' to x' + y. The rest, i.e., multiplying by z and using distributivity, is trivial. The nontrivial part is what I showed in post #3. In fact, your transformation is shorter because you used distributivity of addition over multiplication, which allows changing xy + x' to (x' + x)(x' + y). So, good job!