Why (NOT A)(NOT B)(C) + B = (NOT A)(C) + B [Boolean Algebra]

In summary, the step in simplifying (NOT A)(NOT B)(C) + B involves expanding B as (¬A)(C)(B) + B and then combining the first two terms to yield (¬A)(C). This allows for the simplification of the original expression.
  • #1
rehcarlos
8
0

Homework Statement


I'm studying function simplification in boolean algebra, and I didnt understand the following step:
(NOT A)(NOT B)(C) + B = (NOT A)(C) + B

What happened to the NOT B?

Homework Equations


The Attempt at a Solution

 
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  • #2
Expand (¬A)(C) on the right hand side as (¬A)(¬B)(C)+(¬A)(B)(C) and simplify.
 
  • #3
Hey DH thanks for helping,

but I still don't get it, I mean...

I need to know what's the next step of (NOT A)(NOT B)(C) + B. In your answer, you are saying that I need to expand the right side... but in a real exercise, I wouldn't know what the right side would look like
 
  • #4
On the left hand side, then.

Expand B as B = (any boolean expression whatsoever)B + B. Here we'll use B = (¬A)(C)(B) + B. Then the left hand side becomes (¬A)(¬B)(C) + (¬A)(C)(B) + B. Combine the first two terms and simplify to yield (¬A)(C).
 
  • #5
Got it! Thanks
 
  • #6
test
 

1. Why is the equation (NOT A)(NOT B)(C) + B = (NOT A)(C) + B in Boolean Algebra?

The equation (NOT A)(NOT B)(C) + B = (NOT A)(C) + B is a fundamental rule in Boolean Algebra known as the distributive law. It states that when the same term (B) is added to two different expressions ((NOT A)(NOT B)(C) and (NOT A)(C)), the result will be the same as if the term was factored out and only added once.

2. How is this equation useful in Boolean Algebra?

The distributive law is a key tool in simplifying Boolean expressions. It allows us to break down complex expressions into simpler ones, making it easier to analyze and manipulate logical statements.

3. Can this equation be applied to other logical operations besides NOT and AND?

Yes, the distributive law can be applied to any two logical operations that follow the same properties as NOT and AND. This includes operations like OR, NAND, and NOR.

4. What happens if the equation is written as (NOT A)(NOT B)(C) + B = (NOT A)(C)?

In this case, the equation would still be valid and true. The term B would simply drop out of the expression, as it is being added to both sides.

5. Is there a geometric interpretation of this equation?

Yes, the distributive law can also be interpreted geometrically using Venn diagrams. The left side of the equation represents the area where the three sets (NOT A), (NOT B), and (C) overlap, plus the area of set B. The right side of the equation represents the area where only sets (NOT A) and (C) overlap, plus the area of set B. These two areas are equal, illustrating the distributive property.

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