Which is logically equivalent to ~p -> q?
a.) ~q -> p
b.) ~q -> ~p
c.) q -> p
d.) q -> ~p
I am not certain of your usage of ->, but I have a technique you can use that will work always.
Construct a truth table, such as this:
http://www.gateways2learning.com/SetTheory/EquivalentStatements.pdf
It is just like your previous post. The contrapositive of
~p -> q
reverses the direction of implication, and the logical values:
~(q) -> ~(~p)
~q -> p
To determine which statement is logically equivalent to ~p -> q, we can rewrite the statement using logical equivalences to see which option matches the result.
~p -> q can be rewritten as ~(~p) ∨ q using the logical equivalence of → (implication) operator, which states that p -> q is equivalent to ~p ∨ q.
Applying De Morgan's law, we can simplify ~(~p) to p, so now the statement becomes p ∨ q.
Option A: ~q -> p can be written as ~(~q) ∨ p, which simplifies to q ∨ p. This does not match p ∨ q, so option A is not the correct answer.
Option B: ~q -> ~p can be written as (~q) ∨ (~p), which is equivalent to q ∨ ~p. This does not match p ∨ q, so option B is not the correct answer.
Option C: q -> p remains the same and matches p ∨ q, so option C could be logically equivalent to ~p -> q.
Option D: q -> ~p can be written as q -> (~p), which is equivalent to ~q ∨ ~p. This does not match p ∨ q, so option D is not the correct answer.
After analyzing the logical equivalences, we can conclude that option C, q -> p, is logically equivalent to ~p -> q.