To express the set [tex]\( \{x \mid x \in \mathbb{N} \text{ and } 15 < x \leq 20\} \)[/tex] using the roster method, we need to list all the natural numbers that satisfy the conditions [tex]\( 15 < x \leq 20 \)[/tex].
First, let's identify the natural numbers greater than 15 and less than or equal to 20. The natural numbers begin from 1 and go up in increments of 1 (i.e., 1, 2, 3, 4, ...).
The boundary defined by [tex]\( 15 < x \)[/tex] means that [tex]\( x \)[/tex] must be greater than 15. Therefore, we start from the next natural number, which is 16.
The boundary defined by [tex]\( x \leq 20 \)[/tex] indicates that [tex]\( 20 \)[/tex] is included in the set.
So, we need to list all natural numbers from 16 up to and including 20. These numbers are:
[tex]\[ \{16, 17, 18, 19, 20\} \][/tex]
Hence, the set [tex]\( \{x \mid x \in \mathbb{N} \text{ and } 15 < x \leq 20\} \)[/tex] expressed using the roster method is:
[tex]\[ \{16, 17, 18, 19, 20\} \][/tex]
