Is IEEE 754 float arithmetic associative, commutative, distributive, etc? Why?
The IEEE 754 standard defines exactly how floating-point arithmetic is performed. For many interesting theorems, you will need to examine the exact definition. For some less interesting ones, like a+b = b+a or ab = ba, all you need to know that IEEE 754 always calculates the exact result, rounded in a deterministic way. For non-theorems, like (a+b)+c = a+(b+c), which is false, all you need is a counterexample which is usually easy to find.