Question

If we show that ¬p → (r ∧¬r) is true for some proposition r, then this is a

Answer 1

To show that ¬p → (r ∧¬r) is true, we need to examine the truth values of both sides of the implication. The right side of the implication is always false, so the statement is vacuously true for any proposition r.

Step-by-step explanation:

If we want to show that ¬p → (r ∧¬r) is true for some proposition r, we have to examine the truth values of both sides of the implication.

• ¬p is the negation of p, so it will be true if p is false and false if p is true.
• (r ∧¬r) is a conjunction of r and ¬r, so it will be true only if both r and ¬r are true, but this is impossible since r and ¬r cannot be simultaneously true.

Since the right side of the implication (r ∧¬r) is always false, the statement ¬p → (r ∧¬r) is vacuously true for any proposition r, regardless of the truth value of p.