Question 21

First, evaluate the cardinality of set $S_1$:
An $n \times n$ matrix has $n^2$ entries. Each entry can be chosen from the 3-element set $\{a, b, c\}$. Therefore:

Next, evaluate the cardinality of set $S_2$:
Set $S_2$ consists of all functions from the domain $\{0, 1, 2, \dots, n^2 - 1\}$ (which has $n^2$ elements) to the codomain $\{0, 1, 2\}$ (which has 3 elements). The number of such functions is:

Because both $S_1$ and $S_2$ are finite sets with exactly the same number of elements ($|S_1| = |S_2| = 3^{n^2}$):

• There exists a bijection from $S_1$ to $S_2$ (making option C correct, and option A incorrect).
• Any bijection between two finite sets of identical size is both an injection and a surjection. Therefore, there exists a surjection from $S_1$ to $S_2$ (making option B correct).
• Since bijections are injective, injections exist between $S_1$ and $S_2$, so option D is incorrect.