Answer
If all the variables, p, q and r are True:
- every disjunction is True
- then every conjunction of them is True.
If all the variables are False:
- every disjunction is also True, because there's always a variable with a negation in it.
- So the conjunction of them is also True.
Work Step by Step
1) If all the variables are True. We have p= T, q =T and r = T
2) Plug the value of p, q and r into the statement:
(True OR [not] True) AND (True OR [not] True) AND (True OR [not] True)
3) [not] True becomes False
(True OR False) AND (True OR False) AND (True OR False)
4) True OR False = True, and True AND True = True
Now let's do the case where all variables are False.
2) Plug the value of p, q and r into the statement:
(False OR [not] False) AND (False OR [not] False) AND (False OR [not] False)
3) [not] False becomes True
(False OR True) AND (False OR True) AND (False OR True)
4) True OR False = True, and True AND True = True
