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.

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?

Re: Need help with associative law with 4 variables

Thee two expressions are NOT equivalent.

$\displaystyle [P^{(Q^R)}]^S \ne (P^Q)^{(R^S)}$

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)?

Re: Need help with associative law with 4 variables

Quote:

Originally Posted by

**ebaines** Thee two expressions are NOT equivalent.

$\displaystyle [P^{(Q^R)}]^S \ne (P^Q)^{(R^S)}$

My guess is that ^ in the OP denotes conjunction and not exponentiation.

Quote:

Originally Posted by

**lrpronesti** 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.

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)]?

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

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

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?