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Question

Show that: (r → ~ q, r ∪ S, S → ~ q, p → q) ↔ ~ p

are inconsistent.

Comments

By contradiction: Assume that ~ p is true. Since p → q is true, this means that q must be true as well. However, since S → ~ q is true and S ∪ r is also true, this implies that r → ~ q is true, which means that q must be false. This is a contradiction, indicating that the given set of statements are inconsistent.

Answer 1

To show that the set of statements (r → ~ q, r ∪ S, S → ~ q, p → q) ↔ ~ p is inconsistent, we need to find a way to derive a contradiction from the statements.

One way to do this is to use the first three statements to derive the conclusion that q is always false, and then use the fourth statement to derive the conclusion that q is true. This creates a contradiction, as q cannot be both true and false at the same time.

Here is one possible proof of the inconsistency:

1. r → ~ q (given)
2. r ∪ S (given)
3. S → ~ q (given)
4. r ∪ S = r (by definition of union)
5. ~q (by 1, 4 modus ponens)
6. p → q (given)
7. ~p (by 5, 6 contrapositive)
8. ~p and q (by 5, 7)

Step 1-3 show that q is always false. Step 6-8 show that q is true, which is contradictory. Therefore, the set of statements (r → ~ q, r ∪ S, S → ~ q, p → q) ↔ ~ p is inconsistent.