Using the truth table, show that the following proposition is a contradiction.
negation of [((p → negation of q) ∧ q) → negation of p]
so, did you build the truth table? What did you get?
(p^q)^~q is contradiction
To determine whether a proposition is a contradiction or not, we need to construct its truth table and verify if there is any row where the proposition is true.
Let's break down the given proposition step by step:
1. negation of [((p → negation of q) ∧ q) → negation of p]
2. To simplify this proposition, let's focus on the innermost part: (p → negation of q)
3. The negation of q is represented as ¬q.
4. So, now we have: (p → ¬q)
5. Next, let's combine this with q using the conjunction operator ∧: ((p → ¬q) ∧ q)
6. Let's consider the final part of the proposition: ((p → ¬q) ∧ q) → negation of p
7. Finally, we negate p as ¬p: ((p → ¬q) ∧ q) → ¬p
Now, we have a complete proposition. Let's construct its truth table:
First, create columns for p, q, ¬q, (p → ¬q), ((p → ¬q) ∧ q), ¬p, and the final proposition.
| p | q | ¬q | (p → ¬q) | ((p → ¬q) ∧ q) | ¬p | ((p → ¬q) ∧ q) → ¬p |
|---|---|----|----------|----------------|----|---------------------|
| T | T | F | F | F | F | T |
| T | F | T | T | F | F | T |
| F | T | F | T | T | T | F |
| F | F | T | T | F | T | T |
Checking each row, we can see that there is no row where the final proposition is true. Hence, the given proposition is a contradiction since it is always false, regardless of the truth values assigned to p and q.