Why Is DeMorgan's Theorem Valid According to Truth Tables?

[XodoX]

1. Demonstrate by means of truth tables the validity of the following identities.

a) DeMorgan's theorem for two variables

X'Y' = X' + Y'

' is the bars that are above the letters.




Can someone explain this to me? I mean the truth table is just going to look like this: X'Y' X' Y'
I just need to know where and 0's and 1's go and why. Seems so easy, but I haven't been able to understand it.

 

[berkeman]

1. Demonstrate by means of truth tables the validity of the following identities.

a) DeMorgan's theorem for two variables

X'Y' = X' + Y'

' is the bars that are above the letters.




Can someone explain this to me? I mean the truth table is just going to look like this: X'Y' X' Y'
I just need to know where and 0's and 1's go and why. Seems so easy, but I haven't been able to understand it.

Your table will have 4 coulmns. Fist is X, second is Y, third is X'Y' and forth is X' + Y'.

Well, except that I think you forgot something in your statement of DeMorgan's Law.

EDIT -- it's just a typo in how you wrote it out...

 

[XodoX]

Your table will have 4 coulmns. Fist is X, second is Y, third is X'Y' and forth is X' + Y'.

Well, except that I think you forgot something in your statement of DeMorgan's Law.

EDIT -- it's just a typo in how you wrote it out...

Thanks. I just thought it was 3. How do I find out about the 0's and 1's ? I know DeMorgan's but don't know how to apply it to this one.

 

[berkeman]

Thanks. I just thought it was 3. How do I find out about the 0's and 1's ? I know DeMorgan's but don't know how to apply it to this one.

(hint -- check out the wikipedia.org page on DeMorgan's Law...)

 

[DeeNos]

Truth tables are a tool used in logic and mathematics to help determine the validity of statements or expressions. They are particularly useful in proving identities, such as DeMorgan's theorem for two variables.

To understand how truth tables work, let's first define some terms. In a truth table, the variables (in this case, X and Y) represent statements or propositions that can either be true (1) or false (0). The operations being performed, represented by the symbols ' (not) and + (or), are logical operators that combine these statements.

Now, let's break down the expression X'Y' = X' + Y'. This is saying that the negation (') of X AND the negation of Y are equivalent to the negation of X OR Y. In other words, if X and Y are both false, then the negation of both X and Y is also false. Similarly, if either X or Y (or both) are false, then the negation of either X or Y (or both) is also false.

To prove this using a truth table, we need to exhaust all possible combinations of X and Y and see if the statement holds true for each one. In this case, there are four possible combinations:

X Y X' Y' X'Y' X' + Y'
0 0 1 1 1 1
0 1 1 0 0 0
1 0 0 1 0 0
1 1 0 0 0 0

As you can see, in each row, the value for X'Y' is the same as the value for X' + Y'. This proves that the two sides of the equation are equivalent, and therefore DeMorgan's theorem for two variables is valid.

In summary, truth tables are a way to systematically test all possible combinations of statements and operators to determine the validity of an expression. They may seem confusing at first, but with practice and understanding of the underlying logic, they can be a powerful tool in proving mathematical identities.