Answer: Therefore, the statement is true.
Explanation:
The given statement:
(p ^ q ^ r) v (p ^ q ^ ~r) v (p ^ ~q ^ r) = (p ^ q) v (p ^ r)
is true in classical logic. This is known as the Distributive Law of Disjunction over Conjunction, which is a valid logical equivalence.
Here's a brief explanation:
Left-hand side (LHS):
(p ^ q ^ r) v (p ^ q ^ ~r) v (p ^ ~q ^ r) represents a disjunction (OR) of three different conjunctions (AND).
Each conjunction has p as a common factor.
Right-hand side (RHS):
(p ^ q) v (p ^ r) represents a disjunction (OR) of two conjunctions (AND).
Here, p is also a common factor.
In both the LHS and RHS, the common factor p is combined with different conditions using conjunctions (AND). Since both sides have the same common factor p and use the same logical connectives, they are equivalent. Therefore, the statement is true.