Boolean algebra theorem question

[bcjochim07]

Homework Statement

My book contains this boolean algebra theorem:

A + notA * B = A + B

I have verified the validity of this statement using truth tables, but I find that I am unable to derive it. Our professor gave us a few steps to simplify Boolean expressions:

1) change all variables to their complements
2) change all ORs to ANDS and all ANDS to ORS simultaneously
3) take the complement of the entire expression

Homework Equations

The Attempt at a Solution

When I try to follow these steps, here's what happens:

step 1) A + notA * B => notA + A * notB
step 2) notA + A * notB => notA * A + notB
step 3) notA * A + notB +=> not(notA * A + notB)

and then breaking the longest bar using one of DeMorgan's theorems:

not(notA*A+notB) = not(notA*A) * not(notB) = not(0) * B = 1 * B = B

Maybe I'm making this more complicated than it is? What am doing wrong?

Thanks.

 

[Zayer]

No need for all this steps

You can do it in two steps

1) X+YZ = (X+Y)(X+Z)
2) X + X' = 1

 

[Mene]

I can understand your confusion. Boolean algebra can be tricky, but it is a powerful tool for simplifying logical expressions and making them easier to understand. Let's break down the steps you have taken and see where the error may lie.

Step 1: In this step, you correctly changed all variables to their complements. However, you have also changed the AND and OR operations. Remember, in Boolean algebra, the AND operation is represented by a dot (.) and the OR operation is represented by a plus (+). So, the correct expression after this step would be: notA + not(notA) . B.

Step 2: Here, you have correctly changed all ORs to ANDs and vice versa. But, you have also incorrectly applied the distributive property. The correct expression after this step would be: notA . B + not(notA) . B.

Step 3: In this step, you have correctly taken the complement of the entire expression. However, the expression you have written is incorrect. The correct expression after this step would be: not(notA . B + not(notA) . B).

Now, we can apply DeMorgan's theorem to simplify this expression. According to DeMorgan's theorem, the complement of a AND operation is the OR operation of the complements of the operands, and the complement of a OR operation is the AND operation of the complements of the operands. So, the final expression would be: not(notA) + notB.

Simplifying this further, we get: A + B. Therefore, we have successfully derived the boolean algebra theorem: A + notA . B = A + B.

I hope this explanation helps you understand the steps and the error you made. Remember, practice makes perfect, so keep practicing and you will become more comfortable with boolean algebra.