1962 AHSME Problem 27

Let a @ b represent the operation on two numbers, a and b, which selects the larger of the two numbers, with a@a = a. Let a ! b represent the operator which selects the smaller of the two numbers, with a ! a = a. Which of the following three rules is (are) correct?

\begin{array}{l} \textbf{(1)}\ a@b = b@a \qquad \\ \textbf{(2)}\ a@(b@c) = (a@b)@c \qquad \\ \textbf{(3)}\ a ! (b@c) = (a ! b) @ (a ! c) \end{array}

\textbf{(A)}\ (1)\text{ only} \qquad \textbf{(B)}\ (2) \text{ only} \qquad \textbf{(C)}\ \text{(1) and (2) only}\qquad \textbf{(D)}\ \text{(1) and (3) only}\qquad \textbf{(E)}\ \text{all three}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.