What is the difference between these two Boolean expressions?

Question

I’m checking the truth table on this website on the following expressions:

But each one of them gives me a different way out:

Answer 1

It is a matter of operations precedence, as occurs in mathematics (multiplication and division always occur before addition and subtraction unless there is something grouping them together, such as the use of parentheses), or in every expression in codes we write on the computer, I think I can talk like this because you already know this.

The image is clear to those who know the subject well, but the tables that are in the question make it clearer even to those who do not know it that is very different.

The point is a logical operator AND (I’ve talked about it in code) which says that for the result to be true the two operands must be true, the rest always gives false. The apostrophe denies his operand (NOT), that is, what is true turns false and vice versa. The parentheses group the operation and what is inside must be accomplished before what is outside, so what is outside will operate as a result of all that is inside.

So the first table takes B and denies it, after all the negation operator is next to it. It has priority over the other operator. C is also denied. With the two values denied proceed the relation of E. The first expression has 0 and 0, then both are denied individually, and remain 1, there 1 and 1 gives 1. In the second line and the other is done the same, but as always at least one of them is 0, the E requires that both are 1 to give 1, then the result is 0.

The second table changes because of parentheses. B is related to C through the operation of E and this result is denied by the operator that is valid for all value resulting within the parentheses. In the first three lines there is a 0, then the result will be 0 because E requires both to be 1 to give 1, then 0 there is a negation and the final result is 1. In the last line both are 1, then gives 1 and denied the ending gives 0.

Tables are inverted in relation to the image.

Answer 2

After reading some books, I am answering my own question in more detail and emphasizing what is really important and what was the purpose of the question..

Following the Truth Table above it is possible to notice the difference between the gates logic and most importantly, which is the "type of dam symbology" of the entrances".

Using these entries as an example:

We can observe that the exits from is different from , by the following question:

• To first boolean expression is blocking the entrances A and B
• To second boolean expression is blocking the exits of the logic gate AND of the entries A and B. Knowing this with the "symbology of this dam", we must apply the De Morgan’s theorem.

Note

With the application of the theorem in the second expression we can observe that in fact it is: (so much so that this same boolean expression has the same output as the boolean expression ).

The same goes for the boolean expression , when the quoted theorem is applied this expression is equivalent to: .