Answer
This is a compound proposition made up of two individual propositions; the first proposition is (P v Q v R), and the second is (¬P v ¬Q v ¬R).
P v Q v R only evaluates to false when P,Q,R are all FALSE
!P v !Q v !R only evaluates to false when are all TRUE
So, if P,Q,R are all the same, then one of these propositions will
be false.
And since we know that TRUE AND FALSE = FALSE, and FALSE AND TRUE = FALSE, then P,Q,R all being the same value will result in the compound proposition being FALSE.
If at least one differs from the other, then both propositions will
become true, P v Q v R evaluates to true when at least one is true,
and !P v !Q v !R evaluates to true when at least one is false
Work Step by Step
Let's use an example, with P = T(rue), Q = F(alse), and R= F(alse). Let's plug them into the compound proposition.
(P v Q v R) = (T v F v F) = T.
(¬P v ¬Q v ¬R) = (¬T v ¬F v ¬F) = (F v T v T) = T.
(P v Q v R) ^ (¬P v ¬Q v ¬R) = (T) ^ (T) = T.
Let's use another example using the same truth values, with P= T(rue), Q = T(rue), R = T(rue)
(P v Q v R) = (T v T v T) = T.
(¬P v ¬Q v ¬R) = (¬T v ¬T v ¬T) = (F v F v F) = F.
(P v Q v R) ^ (¬P v ¬Q v ¬R) = (T) ^ (F) = F.
