Question

Let \( X = \{ x \in S_4 : x^3 = \text{id} \} \) and \( Y = \{ x \in S_4 : x^2 \neq \text{id} \} \). If \( m \) and \( n \) denote the number of elements in \( X \) and \( Y \), respectively, then which one of the following is TRUE?

• A. \( m \) is even and \( n \) is even
• B. \( m \) is odd and \( n \) is even
• C. \( m \) is even and \( n \) is odd
• D. \( m \) is odd and \( n \) is odd

Hint:

To find the number of elements in a set defined by conditions, list all possible elements and apply the constraints. Count the elements that satisfy the condition.

Answer 1

The Correct Option is B

Step 1: Elements of \( X \).
The set \( X \) consists of elements \( x \in S_4 \) such that \( x^3 = \text{id} \). These are the elements of order 3 in \( S_4 \). The order 3 elements in \( S_4 \) are the 3-cycles, and there are exactly 8 such elements: \[ (1\ 2\ 3), (1\ 3\ 2), (1\ 2\ 4), (1\ 4\ 2), (1\ 3\ 4), (1\ 4\ 3), (2\ 3\ 4), (2\ 4\ 3) \] Thus, \( m = 8 \), which is even. 
Step 2: Elements of \( Y \).
The set \( Y \) consists of elements \( x \in S_4 \) such that \( x^2 \neq \text{id} \). This means \( Y \) includes all elements of \( S_4 \) except for the identity and the 2-cycles (whose square is the identity). The total number of elements in \( S_4 \) is 24, and the identity and the 2-cycles (6 elements) must be excluded: \[ n = 24 - 6 = 18 \] Thus, \( n = 18 \), which is even. 
Final Answer: \[ \boxed{m \text{ is odd and } n \text{ is even}} \]