Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement.
~[(~p ^ q) ^ r]
~p = F
~p ^ ~q = F
(~p ^ q) ^ r = F
~[(~p ^ q) ^ r] = T
To find the truth value of the compound statement ~[(~p ^ q) ^ r], we need to determine the truth values of the individual propositions and apply the logical operations.
First, let's determine the truth value of p, q, and r:
- p represents a true statement.
- q represents a false statement.
- r represents a false statement.
Now, let's break down the compound statement and apply the logical operations step by step:
1. ~p represents the negation of p. Since p is true, ~p is false.
2. ~p ^ q represents the conjunction (logical AND) of ~p and q. Since ~p is false and q is false, ~p ^ q is false.
3. (~p ^ q) ^ r represents the conjunction (logical AND) of (~p ^ q) and r. Since (~p ^ q) is false and r is false, (~p ^ q) ^ r is false.
4. Finally, ~[(~p ^ q) ^ r] represents the negation of (~p ^ q) ^ r. Since (~p ^ q) ^ r is false, ~[(~p ^ q) ^ r] is true.
Therefore, the compound statement ~[(~p ^ q) ^ r] is true.