Simple TRUTH TABLES questions - Don't answer if you don't want to help!?

Question

PLEASE!!!!!!!!!!! don't answere if you can't help!!!!!! THANKS
For those who are willing to help, THANK YOU!!!!!!!!!!

Use truth tables to establish the following logical equivalencies known as the distributive laws.


1.P v (Q ^ R) ≡ (P v Q) ^ (P v R)
2.P ^ (Q v R) ≡ (P^Q) v (P^R)

Answer 1

no idea, but I hope someone helps you

Answer 2

Well, this is kind of hard to show in this format, but here goes :) If you can't see my tables properly in the Yahoo font, you might want to copy it and paste it into Notepad or in MS Word with Courier New font or something so that everything lines up.

You need to show what each of the individual values P, Q, and R can be, then show what that would give for the parts in parentheses, and then put together the whole thing on a side of the congruence symbol. All of the results should match on both sides of the congruence symbol for all possibilities.

1) Notice that the results for the last 2 columns are the same, so for all possible values of P, Q, and R, the congruence is proven...

P | Q | R | Q^R | PVQ | PVR | PV(Q^R) | (PVQ)^(PVR)
---------------------------------------------------
T | T | T | T | T | T | T | T
T | T | F | F | T | T | T | T
T | F | T | F | T | T | T | T
T | F | F | F | T | T | T | T

F | T | T | T | T | T | T | T
F | T | F | F | T | F | F | F
F | F | T | F | F | T | F | F
F | F | F | F | F | F | F | F

2) Again, note that the results for the last 2 columns are the same, showing the congruence for all possible values of P, Q, and R.

P | Q | R | QVR | P^Q | P^R | P^(QVR) | (P^Q)V(P^R)
---------------------------------------------------
T | T | T | T | T | T | T | T
T | T | F | T | T | F | T | T
T | F | T | T | F | T | T | T
T | F | F | F | F | F | F | F

F | T | T | T | F | F | F | F
F | T | F | T | F | F | F | F
F | F | T | T | F | F | F | F
F | F | F | F | F | F | F | F

Answer 3

Here I just do problem 1 as an example. Hopefully, you can follow the example to do problem 2.

First, you need to know how many rows you need to draw. Since you have three logical variables P, Q and R, there are 2^3 = 8 possible different combinations.

Second, you put the last two columns for P v (Q ^ R) and (P v Q) ^ (P v R). This way it is easy to see whether they have the same truth values.


P..Q..R...Q^R..P v Q..(P v R)..P v (Q ^ R) ..(P v Q) ^ (P v R)
T..T..T..T..T..T..T..T
T..T..F..F..T..T..T..T
T..F..T..F..T..T..T..T
T..F..F..F..T..T..T..T
F..T..T..T..T..T..T..T
F..T..F..F..T..F..F..F
F..F..T..F..F..T..F..F
F..F..F..F..F..F..F..F

The data in the table show that P v (Q ^ R) and (P v Q) ^ (P v R) have exactly the same truth values for all possible combinations. Therefore, P v (Q ^ R) ≡ (P v Q) ^ (P v R) .

End of proof.

Answer 4

Look at this link for the first one:

http://i2.photobucket.com/albums/y16/zorro1267/untitled.jpg

I am too lazy to do the 2nd one, and it would be good practice for you anyways. Just follow my example.