Question

For each of these compound propositions, use the conditional-disjunction equivalence (Example 3) to find an equivalent compound proposition that does not involve conditionals.

a) ¬p → ¬q
b) (p ∨ q) → ¬p
c) (p → ¬q) → (¬p → q)

Answer 1

To convert propositions with conditionals to ones without, we can use the conditional-disjunction equivalence. For example, ¬p → ¬q is equivalent to p ∨ ¬q, (p ∨ q) → ¬p simplifies to ¬p ∨ ¬q, and (p → ¬q) → (¬p → q) simplifies to (p ∨ q).

Step-by-step explanation:

To convert compound propositions that involve conditionals into ones that do not, we use the conditional-disjunction equivalence. The equivalence states that a conditional statement p → q is logically equivalent to its contrapositive ¬q → ¬p, and also to its disjunction ¬p ∨ q. This means that 'if p then q' is equivalent to saying 'not p or q'.

Let's apply this to the compound propositions given:

1. ¬p → ¬q is equivalent to p ∨ ¬q
2. (p ∨ q) → ¬p is equivalent to ¬(p ∨ q) ∨ ¬p, which simplifies to ¬p ∨ ¬q
3. For (p → ¬q) → (¬p → q), we convert each conditional separately, resulting in ¬(p → ¬q) ∨ (¬p → q), which simplifies to (p ∨ q) ∨ (¬¬p ∨ q), and further simplifies to (p ∨ q)