Final answer:
To determine if (¬p∧(p→q))→¬q is a tautology, we break down the statement and create a truth table to evaluate all possibilities. If it is true for all scenarios, it is a tautology; otherwise, it is not.
Step-by-step explanation:
To determine whether (¬p∧(p→q))→¬q is a tautology, we'll analyze the logical statement step by step using a truth table.
Understanding the Logical Connectives
Firstly, we need to understand the logical connectives in the statement:
¬: Not (negation)
∧: And (conjunction)
→: Implies
Breaking Down the Compound Statement
The compound statement can be broken down into simpler components:
¬p (not p)
p → q (p implies q)
¬p∧(p→q) (not p and p implies q)
(¬p∧(p→q))→¬q (if not p and p implies q, then not q)
Analyzing with a Truth Table
We then create a truth table to exhaust all possible truth values for p and q and determine the truth value of the whole statement for each combination. If the statement is true in all possible scenarios, then it is a tautology.
If we find that (¬p∧(p→q))→¬q is true in all cases, then we have a tautology. However, if there is even one case where the statement is false, then it is not a tautology.