15-31 Let $a(\mathcal{D}) b$ denote the operation of selecting the larger of the two numbers $a$ and $b$, and $a(1) a = a$. Also, let $a(S) b$ denote the operation of selecting the smaller of the two numbers $a$ and $b$, and $a(S) a = a$.
The following three conclusions:
(1) $a(1) b = b(\mathbb{D}) a$.
(2) $a(\mathcal{D})(b \mathbb{D}) c = (a(\mathbb{D}) b)(\mathbb{D}) c$.
(3) $a(S)(b(\mathbb{D}) c) = (a(S) b)(\mathbb{D})(a(S) c)$.

Which of the following is correct?
(A) Only (1).
(B) Only (2).
(C) Only (1) and (2).
(D) Only (1) and (3).
(E) All.
(13th American High School Mathematics Examination, 1962)
[Solution] From the given conclusion (1) is obviously true.
For (2), selecting the larger one between $b, c$ and then selecting the larger one with $a$ yields the same result as selecting the larger one between $a, b$ and then selecting the larger one with $c$ (both results are the maximum of $a, b, c$).
For (3), first assume $a$ is the smallest among $a, b, c$. In this case, $a(S(b(D) c)=a$, and
$$
(a \text { (S) } b)(D)(a) c)=a \text {, }
$$

so the conclusion holds in this case;
If $b$ or $c$ is the smallest among $a, b, c$, for example, $b$, then
$$
a(S)(b(\mathbb{D} c)=a(S) c \text {, }
$$

and $(a(\$) \mathbb{D}(a(s)=b(1)(a(\$)=a(\$) c$ (note that $b$ is the smallest). In summary, (1), (2), and (3) all hold.
Therefore, the answer is $(E)$.