Proposition Logic problem
Anyone good with proposition logics? that can help out?
im stuck on one question and can't figure out the next step.
The question states : Use the rules of inference to prove the following.
(p -> q) ^ (r -> s) ^ [t -> ~(q V s) ^ t => (~p ^ ~r)
note : the " ~ " sign means NOT,
the " ^ " sign means AND,
the " -> " sign means implies/conditional
the " V " sign means OR
the " => " sign means resultant or conclusion is.
I've set up my working up to look like this.
P -> q H1(Hypothesis 1)
r -> s H2(Hypothesis 2)
t -> ~(q V s) H3(Hypothesis 3)
t H4(Hypothesis 4)
~p ^ ~r C (Conclusion)
Step one. I got H4 with H3 and AND both of them >> H4 ^ H3 which gives:
t ^ [t -> ~(q V s)] and since t ^ t cancels out with the modus ponens rules it leaves us with ~(q V s).
So now that H4 and H3 is gone we have H1 and H2 left. This is where i got stuck.
I decided to use de morgan's law to simplify which gave me >> ~(q V s) => ~q ^ ~s and we can call this theory one (1).
So far i've only got up to this part and now i'm stuck could you help me figured out what i need to do next?
Working : t ^ [t -> ~(q V s)] <=> ~(q V s) H4 ^ H3
<=> ~q ^ ~s (1).
Next step = ? please help