What Is the Output of F = A + B.C in Boolean Algebra?

[jackson6612]

I'm new to this math world, so please explain your reply in as much detail as possible. Thank you.

Please have a look on this link (Example 6.4):
http://img84.imageshack.us/img84/3667/img0023hg.jpg

Boolean function is $$F=A+\overset{\_\_}{B}.C$$. I don't understand even the first step. I don't understand what it means by saying that the function has three variables A, B, and C. The first term A is missing two variables (B and C).

Please help me. Thank you for your time.

 

[mathman]

The first step is simply saying any set (A) can br expressed as the intersection of itself and the whole space. Furthermore the union of any set (B) and its complement (B') = the whole space. Putting this together and you get A=A.(B + B')=A.B + A.B'

. means intersection, + means union.

 

[jackson6612]

Thank you, Mathman.

But these things 'intersection' and 'uniion' are studied under topics of sets. That Boolean function is part of Boolean algebra involving logic gates. So, could you please deal it that way? Further, I don't even get what the question is asking. Could you please shine a light on this too? Thank you very much for all the guidance and your time.

 

[mathman]

The algebra of "logic gates" and the algebra of sets are essentially identical. I happen to be used to sets, so I express it that way. union is equivalent to "or", intersection is equivalent to "and" and complement is equivalent to "not".

As for

The first term A is missing two variables (B and C).

, all the author is saying you can always throw in [B or (not B)] without changing anything.

 

[CellCoree]

Hello, and welcome to the world of Boolean functions! Don't worry, it can seem confusing at first, but with a little explanation, you'll understand it in no time.

First, let's break down the function F = A+B.C. A Boolean function is a mathematical function that takes in Boolean variables (variables that can only have two values, typically 0 or 1) and produces a Boolean output. In this case, A, B, and C are the Boolean variables. They can each have a value of either 0 or 1.

Now, let's look at the function itself. The "+" sign represents the logical OR operation, while the "." represents the logical AND operation. So, the function F = A+B.C can be read as "F equals A OR (B AND C)." This means that the output of the function will be 1 if either A is 1 or both B and C are 1. Otherwise, the output will be 0.

Next, let's look at the example provided in the link. In this example, we are given a truth table, which is a table that shows all possible combinations of inputs and their corresponding outputs for a given Boolean function. This is a helpful tool for understanding how a Boolean function works.

In the first column, we have the inputs A, B, and C. As mentioned before, these can each have a value of either 0 or 1. In the second column, we have the output of the function, which is denoted as F. As you can see, the output of the function changes depending on the inputs.

For example, in the first row, A=0, B=0, and C=0. Plugging these values into the function F = A+B.C, we get F = 0+0.0 = 0. So, when A=0, B=0, and C=0, the output of the function is 0. You can follow the same process for the rest of the rows to understand how the output changes with different input combinations.

I hope this explanation helps you understand Boolean functions and their truth tables a little better. Just remember that A, B, and C are the inputs, and the function F = A+B.C tells us how these inputs will affect the output. Keep practicing and you'll become a pro in no time!