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Math Help - Need help with associative law with 4 variables

  1. #1
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    Need help with associative law with 4 variables

    Use associative laws to show that [P ^(Q^R)]^S is equivalent to (P ^ Q) ^ (R ^ S).

    I know you can switch anything inside the parentheses with anything outside so I got [R ^(Q^P)]^S but I don't know how to go about getting the S into the parentheses to make it look like (P ^ Q) ^ (R ^ S), or are you just allowed to do that because of the associative law?
    Any help would be greatly appreciated.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Need help with associative law with 4 variables

    Hi Irpronesti!

    The associative law allows you to shift parenthesis around.
    It does not allow you to swap operands.

    Inside the square parentheses you can shift the round parentheses.
    This gives you the leftmost part of what you are trying to find.
    How else can you shift the parentheses?
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  3. #3
    MHF Contributor ebaines's Avatar
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    Re: Need help with associative law with 4 variables

    Thee two expressions are NOT equivalent.

    [P^{(Q^R)}]^S \ne (P^Q)^{(R^S)}
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  4. #4
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    Re: Need help with associative law with 4 variables

    Could I move the R outside the parentheses to get R ^ (P ^ Q) ^ S which is equivalent to (R^S)^(P^Q)?
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  5. #5
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    Re: Need help with associative law with 4 variables

    Quote Originally Posted by ebaines View Post
    Thee two expressions are NOT equivalent.

    [P^{(Q^R)}]^S \ne (P^Q)^{(R^S)}
    My guess is that ^ in the OP denotes conjunction and not exponentiation.

    Quote Originally Posted by lrpronesti View Post
    Could I move the R outside the parentheses to get R ^ (P ^ Q) ^ S which is equivalent to (R^S)^(P^Q)?
    You have already been told in post #2 that associativity does not allow changing the order of the operands. The order always has to be P, Q, R, S; only the distribution of parentheses can change. Further, since the problem is about associativity, you are not allowed to omit parentheses. It is not clear whether R ^ (P ^ Q) ^ S means [R ^ (P ^ Q)] ^ S or R ^ [(P ^ Q) ^ S].

    It has also been suggested to apply the law of associativity inside the square brackets. Then [P ^ (Q ^ R)] ^ S becomes [(P ^ Q) ^ R] ^ S. Now one more application of the law is needed to change this expression into the right-hand side.
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  6. #6
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    Re: Need help with associative law with 4 variables

    RIGHT.
    So we go from [P ^(Q^R)]^S to [(P ^ Q) ^ R] ^ S then to [(P ^ Q) ^ (R ^ S)]?
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  7. #7
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    Re: Need help with associative law with 4 variables

    Yes. Do you understand exactly how associativity is applied in both cases? Suppose the law of associativity is

    (A\land B)\land C\Leftrightarrow A\land (B\land C)

    Can you say, for both applications, whether you rewrite left-to-right or right-to-left and what the values of A, B and C are?
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