To write the set [tex]\( \{2, 4, 6\} \)[/tex] in set-builder notation, we need to describe the elements of the set using a property that they all satisfy.
First, note that all elements in the set [tex]\( \{2, 4, 6\} \)[/tex] are:
1. Natural numbers (positive integers).
2. Even numbers.
3. Less than or equal to 6.
Given these observations, the correct notation will combine these properties appropriately.
Let's evaluate each option:
Option A: [tex]\(\{x \mid x \text{ is a natural number less than or equal to 6}\}\)[/tex]
This set includes [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex], which contains odd numbers as well. This is not correct since our set only includes even numbers. Thus, Option A is incorrect.
Option B: [tex]\(\{x \mid x \text{ is an even natural number}\}\)[/tex]
This set includes all even natural numbers such as [tex]\(\{2, 4, 6, 8, 10, \dots\}\)[/tex]. This is not correct because our set only includes numbers up to 6. Thus, Option B is incorrect.
Option C: [tex]\(\{x \mid x \text{ is an even natural number less than or equal to 6}\}\)[/tex]
This description correctly includes only natural numbers that are even and less than or equal to 6. Hence, it matches our set [tex]\(\{2, 4, 6\}\)[/tex] perfectly.
Thus, Option C is correct.
The correct set-builder notation for the set [tex]\( \{2, 4, 6\} \)[/tex] is:
[tex]\[ \{ x \mid x \text{ is an even natural number less than or equal to 6} \} \][/tex]
Therefore, the correct answer is:
Option C. [tex]\(\{x \mid x \text{ is an even natural number less than or equal to 6}\}\)[/tex]
