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Thread: logic circuit formula

  1. #1
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    logic circuit formula

    Given this circuit:



    I need to find a logical formula.

    I came up with this:

    $\displaystyle (\neg (\neg p \wedge q)) \wedge (\neg (\neg p \wedge r) \vee s)))$


    Is this correct? Can you simplify it to this?


    $\displaystyle (p \wedge \neg q) \wedge ((p \wedge \neg r) \vee s))$
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  2. #2
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    We have the following:

    A1 = p' /\ q
    A2 = p' /\ r
    N3 = A1' = p v q'
    N2 = A2' = p v r'
    O1 = N2 v s = p v r' v s

    A3 = N3 /\ O1 = (p v q') /\ (p v r' v s).
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  3. #3
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    Ok, so coming out of N3....if ~(~p $\displaystyle \wedge$ q) = p $\displaystyle \vee$ ~q ??

    A1 = ~p $\displaystyle \wedge$ q
    A2 = ~p $\displaystyle \wedge$ r
    N3 = ~(~p $\displaystyle \wedge$ q) = p $\displaystyle \vee$ ~q
    N2 = ~(~p $\displaystyle \wedge$ r) = p $\displaystyle \vee$ ~r
    O1 = p $\displaystyle \vee$ ~r $\displaystyle \vee$ s
    A3 = p $\displaystyle \vee$ ~q $\displaystyle \wedge$ p $\displaystyle \vee$ ~r $\displaystyle \vee$ s

    (p $\displaystyle \vee$ ~q) $\displaystyle \wedge$ (p $\displaystyle \vee$ ~r $\displaystyle \vee$ s)
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  4. #4
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    Quote Originally Posted by relyt View Post
    Ok, so coming out of N3....if ~(~p $\displaystyle \wedge$ q) = p $\displaystyle \vee$ ~q
    Yes, you negate each proposition and reverse the junction - if it is AND it becomes OR and vice-versa.

    For instance, let p = "Today is sunny" and q = "Today is raining".

    Then, ~p/\q implies: "Today is NOT sunny AND Today is raining".

    So, ~(~p/\q) means: NOT (Today is not sunny and today is raining), or, equivalently "Today is NOT-NOT sunny OR Today is NOT raining). That is, "Today is sunny OR NOT raining". It might be a cloudy weather...
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