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I'm a little confused on when to use brackets in symbolic logic. My initial knowledge was that it was to reduce ambiguity.

I've seen that p ^ (q v r) does not equal (p ^ q) v r via truth table induction.

But I've also proved myself that p ^ q ^ r is logically equivalent to p ^ (q ^ r) via the same logic.

In the textbook I am using the answer for p, q, and r written in English is p ^ q ^ r, but the answer for neither p nor q, but r is (~p ^ ~q) ^ r.

I'm just not quite making the connection on when to use the brackets, and when not too except for some vague patterns that I have noticed.

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Use brackets whenever you find them clarifying, and omit them otherwise (unless you suspect that the reader may need them). Sometimes the use of brackets simply follows from previous steps, and helps keep the reader oriented.

Assuming you do not make any mistakes when placing your brackets, the logic of your argument will be perfectly preserved, so correctness will be all the same.

I assume this is in the context of a class: you should not be marked down for writing $p \land (q \land r)$ as opposed to $p \land q \land r$, as the expressions are equivalent. However, try not to go overboard a la $(((p)) \land ((q) \lor r))$. As a general rule of thumb in your notation: strive to be as clear as possible, and avoid making a mess.

You can think of simplying your logical expressions as similar to simplying fractions in your middle-school algebra homework. Apply similar discretion.

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