# Boolean algebra - distribution

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1. Sep 12, 2016

### Rectifier

The problem
I am trying to show that $a'c' \vee c'd \vee ab'd$ is equivalent to $(a \vee c')(b' \vee c')(a' \vee d)$

The attempt
$(a \vee c')(b' \vee c')(a' \vee d) \\ (c' \vee (ab'))(a' \vee d)$
The following step is the step I am unsure about. I am distributing the left parenthesis over the right.
$(c' \vee (ab'))(a' \vee d) \\ c'a \vee c'd \vee ab'a' \vee ab'd$

I am almost there but the term $ab'a'$ differs. I suspect that you can remove that term since it does not afect the output because it is always false. What do you say about that?

2. Sep 12, 2016

### micromass

Staff Emeritus
Can you show $ab'a' = 0$?

3. Sep 12, 2016

### Rectifier

Unfortunately, I cant, but I know it is true for all values of 1 and 0 for a and b. Since $a$ is the opposite of $a'$ this means that $a \wedge a'$ will always be 0 and it does not matter which value b has . I can write a truth table but I am on my phone so its a bit hard.

4. Sep 12, 2016

### micromass

Staff Emeritus
What are your axioms for Boolean algebra?

5. Sep 12, 2016

### Rectifier

Are you thinking about $x \cdot x' = 0$ ?

6. Sep 12, 2016

### micromass

Staff Emeritus
I don't know, how did you define a boolean algebra?

7. Sep 12, 2016

### Rectifier

In boolean algebra variables can only have two values.

8. Sep 12, 2016

### micromass

Staff Emeritus
Either you tell me how you defined a boolean algebra, or I'm out of this thread.

9. Sep 12, 2016

### Rectifier

I am sorry that my answer didn't fit you but I cant come up with anything better. In any case, I am thankful for the help I have received so far.

10. Sep 12, 2016

### micromass

Staff Emeritus
How are you supposed to prove anything about a Boolean algebra if you're not given its definition or anything??

11. Sep 12, 2016

### Rectifier

12. Sep 12, 2016

### micromass

Staff Emeritus
Those are axioms for a structure that is called a bounded lattice. It's not the axioms for a Boolean algebra. Usually, there are 10 axioms for a Boolean algebra. They can be found here: https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Definition

Note that in this case, it is an axiom that $x\wedge x' = 0$. This is why I asked for your definition, since you might adopt different (but equivalent) ways of doing things.

13. Sep 12, 2016

### Rectifier

I didnt think they were different :). Thank you for you help kind stranger!

So now I can basically remove that ab'a'. Halleluja, I am done with that problen.