How do you represent basic arithmetic using boolean function? [closed]

Say you have an arithmetic problem involving two variables, how do you give a Boolean formula for that using standard techniques so that one gets a minimal formula for a given number of quantifiers?

Consider for example a function that decides if $(1)$ $x>y$ or $(2)$ a function that decides $x|y$?

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• Hint: how do you add in elementary school? – Raphael Dec 7 '14 at 10:59
• Probably this is a stupid question. But curious about quantifier elimination. – T.... Dec 7 '14 at 11:44

I assume you do know how to write a boolean circuit to test whether $x>y$. If not, consult a textbook.
• Thank you. Actually I 'know' does not mean I am comfortable using quantifiers. Is there a reference? I am thinking there is both a $\forall\exists$ and a $\exists\forall$ version. – T.... Dec 7 '14 at 21:34