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Math Help - Proposition Logics - Seriously need help!

  1. #1
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    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)


    Set up

    P -> q ... H1(Hypothesis 1)
    r -> s ... H2(Hypothesis 2)
    t -> ~(q V s) ... H3(Hypothesis 3)
    t ... H4(Hypothesis 4)

    Working:

    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
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  2. #2
    Senior Member
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    That seems correct.

    t ^ [t -> ~(q V s)] => ~q ^ ~s

    (p -> q) ^ ~q => ~p

    (r -> s) ^ ~s => ~r

    conclusion: ~p ^ ~r

    That's what you've done, with lot of details. That doesn't seem weird
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  3. #3
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    yeh i figured it out, i did some uncessary steps, thnx for replying.
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