Let's examine each set to determine whether it is finite or infinite:
### (i) [tex]\( C = \{p \mid p \text{ is a number from 1 to 10 divisible by 2}\} \)[/tex]
To determine whether set [tex]\( C \)[/tex] is finite or infinite, we list all numbers between 1 and 10 that are divisible by 2:
[tex]\[ C = \{2, 4, 6, 8, 10\} \][/tex]
Since there are only 5 elements in [tex]\( C \)[/tex], it is finite.
### (ii) [tex]\( D = \{n \mid n \text{ is a natural number}\} \)[/tex]
Natural numbers [tex]\( n \)[/tex] are the positive integers starting from 1:
[tex]\[ D = \{1, 2, 3, 4, \ldots\} \][/tex]
Since there is no upper limit to natural numbers, the set [tex]\( D \)[/tex] continues indefinitely. Therefore, [tex]\( D \)[/tex] is infinite.
### (iii) [tex]\( B = \{x \mid x \in \mathbb{N} \text{ and } 3x - 1 = 0\} \)[/tex]
To determine the elements of set [tex]\( B \)[/tex], we solve the equation [tex]\( 3x - 1 = 0 \)[/tex]:
[tex]\[ 3x - 1 = 0 \implies 3x = 1 \implies x = \frac{1}{3} \][/tex]
There is no natural number [tex]\( x \)[/tex] that satisfies this equation (as natural numbers are positive integers starting from 1). Hence, the set [tex]\( B \)[/tex] has no elements, and an empty set is finite.
### (iv) [tex]\( A = \{y \mid y < -1, y \text{ is an integer}\} \)[/tex]
We list the integers [tex]\( y \)[/tex] that are less than -1:
[tex]\[ A = \{\ldots, -4, -3, -2\} \][/tex]
Since there is no lower limit to the negative integers, the set [tex]\( A \)[/tex] continues indefinitely in the negative direction. Therefore, [tex]\( A \)[/tex] is infinite.
### Conclusion
Based on the analysis:
1. Set [tex]\( C \)[/tex] is finite.
2. Set [tex]\( D \)[/tex] is infinite.
3. Set [tex]\( B \)[/tex] is finite.
4. Set [tex]\( A \)[/tex] is infinite.
The final classification is as follows:
[tex]\[
(\text{finite}, \text{infinite}, \text{finite}, \text{infinite})
\][/tex]
