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Thread: What should i do next? Logic simplification

  1. #1
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    What should i do next? Logic simplification

    [(p ∨ q) =⇒ r] =⇒ (~q ∧ r)

    ~[~(p ∨ q) v r] v (~q ∧ r) 1.Implication law

    [(p ∨ q) /\ ~r] v (~q ∧ r) 2.Double Negation and DeMorgan law

    I'm stuck at this point.
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  2. #2
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    Re: What should i do next? Logic simplification

    Quote Originally Posted by Dukeham View Post
    [(p ∨ q) =⇒ r] =⇒ (~q ∧ r)
    ~[~(p ∨ q) v r] v (~q ∧ r) 1.Implication law
    [(p ∨ q) /\ ~r] v (~q ∧ r) 2.Double Negation and DeMorgan law
    I'm stuck at this point.
    Look at the truth-table
    What are you trying to simplify?
    From the way it is posted it seems you are proving the implication:
    $[(p \vee q) \Rightarrow r] \Rightarrow (\neg q \wedge r)$
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  3. #3
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    Re: What should i do next? Logic simplification

    I need to simplify the proposition as much as possible so it would be equivalent to the original expression.

    Example:
    [(p ⇒ q) ⇒ (p v q)] 0. Original expression
    [~(~p v q) v (p v q)] 1. Implication law
    [(p /\ ~q) v (p v q)] 2. Doble negation and DeMorgan law
    [(p /\ ~q) vp) v q 3. Associative law
    pvq 4. absorption law
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  4. #4
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    Re: What should i do next? Logic simplification

    Quote Originally Posted by Dukeham View Post
    I need to simplify the proposition as much as possible so it would be equivalent to the original expression.

    Example:
    [(p ⇒ q) ⇒ (p v q)] 0. Original expression
    [~(~p v q) v (p v q)] 1. Implication law
    [(p /\ ~q) v (p v q)] 2. Doble negation and DeMorgan law
    [(p /\ ~q) vp) v q 3. Associative law
    pvq 4. absorption law
    This is hardly a simplification:
    $[(p \vee q) \Rightarrow r] \Rightarrow (\neg q \wedge r)$
    $\begin{gathered} \neg [\neg (p \vee q) \vee r] \vee (\neg q \wedge r) \hfill \\
    [(p \vee q) \wedge \neg r] \vee (\neg q \wedge r) \hfill \\
    [(p \wedge \neg r) \vee (q \wedge \neg r)] \vee (\neg q \wedge r) \hfill \\
    \end{gathered}$

    You can compare this truth-table with this
    to see equivalence.
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  5. #5
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    Re: What should i do next? Logic simplification

    so the last thing I can apply is distributive law?
    If the truth table indicates that they are equivalent, does it mean that the exercise was performed correctly?
    Last edited by Dukeham; Jun 7th 2018 at 05:02 PM.
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  6. #6
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    Re: What should i do next? Logic simplification

    Quote Originally Posted by Dukeham View Post
    so the last thing I can apply is distributive law?
    If the truth table indicates that they are equivalent, does it mean that the exercise was performed correctly?
    $\begin{gathered}
    *~ [(p \vee q) \wedge \neg r] \vee (\neg q \wedge r) \hfill \\
    **~[(p \wedge \neg r) \vee (q \wedge \neg r)] \vee (\neg q \wedge r) \hfill \\
    \end{gathered}$
    Going from * to **distribution is used,
    Last edited by Plato; Jun 7th 2018 at 05:12 PM.
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  7. #7
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    Re: What should i do next? Logic simplification

    Thank you very much. I really appreciate it.
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