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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

If ? is defined as:

which one(s) of the following statements is/are true?

Notice that the expression resulting from the application of the operator is symmetric with respect to the two operands m and n. This means that interchanging them will not change the expression. Therefore, I is true. The operator is commutative.

Also,

while

Therefore,

and IV is also true. The operator is associative.

Applying the operator in a similar manner on the remaining two statements shows that they are not true.