Consider the following sets, where $n \ge 2$: $S_1$: Set of all $n \times n$ matrices with entries from the set $\{a, b, c\}$ $S_2$: Set of all functions from the set $\{0, 1, 2, \dots, n^2 - 1\}$ to the set $\{0, 1, 2\}$ Which of the following choice(s) is/are correct?

Let us determine the cardinalities of both sets $S_1$ and $S_2$:

1. Cardinality of $S_1$: An $n \times n$ matrix contains $n^2$ entries. Each entry can take one of $3$ possible values from the set $\{a, b, c\}$.

   $$|S_1| = 3^{n^2}$$

2. Cardinality of $S_2$: The domain $\{0, 1, 2, \dots, n^2 - 1\}$ has $n^2$ elements and the codomain $\{0, 1, 2\}$ has $3$ elements. The number of functions from a domain of size $n^2$ to a codomain of size $3$ is:

   $$|S_2| = 3^{n^2}$$

Since $|S_1| = |S_2| = 3^{n^2}$ and both sets are finite, there exists a bijection (one-to-one and onto mapping) between $S_1$ and $S_2$.

Because a bijection is both surjective and injective:

• (B) is correct: A surjection exists.

• (C) is correct: A bijection exists.

• (A) and (D) are incorrect because a bijection and an injection both exist.