Proposition Logics - Seriously need help!
Could anyone help me on this problem? i think i did it wrong it looks wierd so i was hoping if someone could show me where i went wrong.
Prove the following using rules of inference.
(p -> q) ^ (r -> s) ^ [t -> ~(q V s) ^ t => (~p ^ ~r)
P -> q ... H1(Hypothesis 1)
r -> s ... H2(Hypothesis 2)
t -> ~(q V s) ... H3(Hypothesis 3)
t ... H4(Hypothesis 4)
t ^ [t -> ~(q V s)] <=> ~(q V s) ... H4 ^ H3 Modus ponens
<=> ~q ^ ~s (1) ... de morgan's law
p -> q => ~q -> ~p (2) ... Contrapositive
r -> s => ~s -> ~r (3) ... Contrapositive
(~q -> ~p) ^ (~q ^ ~s) .... (2) ^ (1)
=> ~p ^ ~s ... Exportation
=> ~s ^ ~p (4) ... Commutativity
(~s -> ~r) ^ (~s ^ ~p) (3) ^ (4)
=> ~r ^ ~p ... Exportation
=> ~p ^ ~r ... Commutativity <--- Conclusion
I might have the rules wrong, could some please check if i've done this correct? i really need to know its urgent. If i have done something wrong could you tell me where i went wrong and if possible could you state the correct rule of inference along with it