# Math Help - Boolean Algebra

1. ## Boolean Algebra

This question seems easy enough but I want to make sure

Q) Use the definition of a Boolean Algebra to give reasons for each step in the proof below:

$\forall a \in B, a \cdot a = a$

Proof:

Let a be any element of B, then

$\begin{array}{cc} a = a \cdot 1 \\=a \cdot (a + \bar{a})\\=(a\cdot a)+(a \cdot \bar{a} \\ =(a \cdot a) + 0 \\ =a \cdot a\end{array}$

first step. $a \cdot 1$ is like saying a and 1

second step $1 = a + \bar{a}$ so we get $a \cdot (a + \bar{a})$

third step is distributive, the + is similar to an 'or' in logic

fourth step $a \cdot \bar{a} = 0$ similar to a set and its complement yield the empty set

fifth step is the left over $a \cdot a$ which is similar to a set intersect itself

2. ## Re: Boolean Algebra

The problem said "Use the definition of Boolean Algebra" and you never stated what that definition is.

3. ## Re: Boolean Algebra

This is the definition I learned:

A Boolean algebra is a set $B$, along with two (binary) operations + and $\cdot$, and a unary operation $a \to \overline{a}$ (called complementation), along with two distinguished elements $0,1 \in B$ such that:

A1) $a + (b + c) = (a + b) + c$, for all $a,b,c \in B$

A2) $a + b = b + a$, for all $a,b \in B$

A3) $a\cdot (b\cdot c) = (a\cdot b)\cdot c$, for all $a,b,c \in B$

A4) $a\cdot b = b\cdot a$ for all $a,b \in B$

A5) $a \cdot (b + c) = (a\cdot b) + (a\cdot c)$, for all $a,b,c \in B$

A6) $a + (b\cdot c) = (a + b)\cdot (a+c)$, for all $a,b,c \in B$

A7) $a + 0 = a$, for all $a \in B$

A8) $a \cdot 1 = a$, for all $a \in B$.

A9) $a + \overline{a} = 1$, for all $a \in B$.

A10) $a \cdot \overline{a} = 0$, for all $a \in B$.

Thus:

$a = a\cdot 1$ (A8)

$= a \cdot (a + \overline{a})$ (A9)

$= (a \cdot a) + (a\cdot\overline{a})$ (A5)

$= (a \cdot a) + 0$ (A10)

$= a \cdot a$ (A7)

Note that EVERY SINGLE STEP is justified by one of the axioms A1-A10, and not by saying, "this is just like in sets, where we have..."

This is what you do in a formal proof, as opposed to an "informal argument" (which may indicate the same line of reasoning).

4. ## Re: Boolean Algebra

I have a list of properties, those properties are what I was told to use as the definition.

5. ## Re: Boolean Algebra

Your job, then, is to see which of my A1-A10 match up with your list of properties, hmm?

6. ## Re: Boolean Algebra

my list of properties is shorter not sure why but yes I will match them up. Next time I will put my more formal version up as opposed to my side work.

7. ## Re: Boolean Algebra

Well, I think I understood your reasoning, but instructors can be nit-picky about these things. They want you to get in the habit of being able to get "more formal" if you need to.

The reason for this is: when we "think" in our heads, we often "skip ahead", due to "having seen this sort of thing before". Sometimes, that can lead to overlooking a minor detail, which possibly might "ruin everything" (like if we divide by 0 in doing calculations).

A formal proof, by contrast, is "air-tight", and there is no arguing with it, unless you want to "abolish the rules of the game" (change the definitions).