p q q r p q r Right now I think it is valid because of De M

p q

q r

¬(p q)

____________

¬r

Right now I think it is valid because of De Morgan\'s law making ¬(p q) into (¬p¬q) and then getting ((¬p¬q) (pq) (q r))-> ¬r... but not really sure where to go from here or how exactly to prove it. Someone said to use a truth table but I don\'t get how the truth table would solve it. So please just explain why the steps taken to solve this are used.

Solution

We are given

p q

q r

¬(p q)

____________

¬r

we can usee rule of inference to solve it

we have

p q

q r

----------------------

p--->r (hypothetical syllogism)

we are also given on eof premises as

¬(p q)

we can also write it as

¬(p q) = ¬p ^ ¬q

or

¬p

¬q

---------

so, we have

p--->r

¬p

¬q

we can use

p--->r

¬p

----------------------------

¬r (modus tollens)........Answer