binary predication

I know basic binary predicate logic. Such as x > y. x or y. x and y.
I've been asked for a class to program four binary predicates that can be used to estabolish a proof. Predicates that are a little more complex.Maybe using integer iterations for example. I can maybe think of one as x = 2y + 1. For all y There exists an x that is odd. etc...
I can program them I just need help with ideas. Thank you.
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