Question

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

• A. There does not exist a bijection from \(S_1\) to \(S_2\).
• B. There exists a surjection from \(S_1\) to \(S_2\).
• C. There exists a bijection from \(S_1\) to \(S_2\).
• D. There does not exist an injection from \(S_1\) to \(S_2\).

Hint:

When two finite sets have equal cardinality, bijections, injections, and surjections between them all exist.

Answer 1

The Correct Option is B, C

To solve this problem, we need to compare the cardinality (the number of elements) of the two sets \( S_1 \) and \( S_2 \) and determine the relationships between them in terms of functions.

1. Set \( S_1 \):
   • \( S_1 \) represents the set of all \( n \times n \) matrices with entries from \(\{a, b, c\}\).
   • The total number of entries in an \( n \times n \) matrix is \( n^2 \).
   • Each entry has 3 choices (either \( a \), \( b \), or \( c \)).
   • Therefore, the number of possible matrices is given by \( 3^{n^2} \). This is because each of the \( n^2 \) entries can independently be one of the 3 choices.
2. Set \( S_2 \):
   • \( S_2 \) is the set of all functions from the set \(\{0, 1, 2, \ldots, n^2-1\}\) to the set \(\{0, 1, 2\}\).
   • Since there are \( n^2 \) elements in the domain and 3 choices for each output value, the total number of functions is also \( 3^{n^2} \).
3. Analysis:
   • Both \( S_1 \) and \( S_2 \) have the same cardinality, \( 3^{n^2} \).
   • Because they have the same number of elements, a bijection (a one-to-one correspondence between the two sets) can exist.
   • This directly implies that both a surjection (onto function) and an injection (one-to-one function) can exist between the two sets. Therefore, a bijection exists.
4. Conclusion:
   • The statement "There exists a bijection from \( S_1 \) to \( S_2 \)" is correct.
   • This also implies that "There exists a surjection from \( S_1 \) to \( S_2 \)" is correct.
   • Statements like "There does not exist a bijection from \( S_1 \) to \( S_2 \)" and "There does not exist an injection from \( S_1 \) to \( S_2 \)" are incorrect since bijections and injections are indeed possible.