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On a homework assignment, we were given

$$x \lt 5y \leftrightarrow x \gt z \rightarrow x + z \geq yw \wedge -x \lt z$$

I broke it up into

$$1~~~x <5y$$ $$2~~~~~x > z$$ $$3~~~x + z \geq yw \wedge -x \lt z$$

The difficulty comes in which order to parse the sections. I don't know how to draw a tree in Latex, but I argued that $\leftrightarrow$ should be a the root and the $\rightarrow$ should be at the first right node. My friends argue that the $\rightarrow$ should be at the root, and the $\leftrightarrow$ should be at the first left node.

My reasoning is that with operators of equal precedence and lack of initial parenthesization, left associativity comes into play, thus rendering $\leftrightarrow$ as the root. My friends argue that parenthesizing the formula results in sections 2 and 3 together, thus making $\rightarrow$ the root.

Who's correct?

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It really depends on how you parenthesize things. If you consider $\leftrightarrow$ and $\rightarrow$ to be of the same precedence, then it depends on whether you assume left or right associativity. You should consult your textbook or teacher for the correct convention. – Tunococ Sep 18 '12 at 0:27
This question isn't really about trees or discrete mathematics, it's about notational precedence and parenthesization. Retagging. (Also best to avoid the word complex in a mathematical context when not talking about complex numbers or simplicial complexes.) – Rahul Sep 18 '12 at 0:29
The parsimony with parentheses yields a glutinous headachy mass of goo. Can we do better here?? – ncmathsadist Sep 18 '12 at 0:29
I second @Tunococ's suggestion. Even with some small amount of internal information (the clauses $x>z\text{ and }-x<z$, for instance), the only way I'd ask this on a homework was if I knew my students had seen enough precedence rules to make this unambiguous. As it stands, there's simply not enough background information to permit a unique answer (and I for one generally hate giving problems with more than one interpretation). – Rick Decker Sep 18 '12 at 2:20

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