To solve the problem described by the set notation [tex]\(\{2a + 1 \mid 1 < a < 8 ; a \in \mathbb{Z}\}\)[/tex], follow these detailed steps:
1. Understand the Constraints on [tex]\(a\)[/tex]:
- The inequality [tex]\(1 < a < 8\)[/tex] specifies the range of integer values that [tex]\(a\)[/tex] can take.
- Since [tex]\(a\)[/tex] must be an integer within this range ([tex]\(\mathbb{Z}\)[/tex] denotes the set of all integers), we identify the possible values for [tex]\(a\)[/tex] which are [tex]\(a = 2, 3, 4, 5, 6, 7\)[/tex].
2. Select the Valid Integer Values for [tex]\(a\)[/tex]:
- To clarify, the possible integer values for [tex]\(a\)[/tex], satisfying [tex]\(1 < a < 8\)[/tex], are:
[tex]\[
a = 2, 3, 4, 5, 6, 7
\][/tex]
3. Apply the Function [tex]\(2a + 1\)[/tex] to Each Valid Value of [tex]\(a\)[/tex]:
- For [tex]\(a = 2\)[/tex]:
[tex]\[
2(2) + 1 = 4 + 1 = 5
\][/tex]
- For [tex]\(a = 3\)[/tex]:
[tex]\[
2(3) + 1 = 6 + 1 = 7
\][/tex]
- For [tex]\(a = 4\)[/tex]:
[tex]\[
2(4) + 1 = 8 + 1 = 9
\][/tex]
- For [tex]\(a = 5\)[/tex]:
[tex]\[
2(5) + 1 = 10 + 1 = 11
\][/tex]
- For [tex]\(a = 6\)[/tex]:
[tex]\[
2(6) + 1 = 12 + 1 = 13
\][/tex]
- For [tex]\(a = 7\)[/tex]:
[tex]\[
2(7) + 1 = 14 + 1 = 15
\][/tex]
4. Compile the Results into a Set:
- Putting all the results together from applying the function [tex]\(2a + 1\)[/tex] yields:
[tex]\[
\{5, 7, 9, 11, 13, 15\}
\][/tex]
5. Finalize and Present the Result:
- Therefore, the final set described by [tex]\(\{2a + 1 \mid 1 < a < 8 ; a \in \mathbb{Z}\}\)[/tex] is:
[tex]\[
\{5, 7, 9, 11, 13, 15\}
\][/tex]
Thus, we have found the set of all possible values of [tex]\(2a + 1\)[/tex] where [tex]\(a\)[/tex] is an integer satisfying [tex]\(1 < a < 8\)[/tex]. The detailed steps show that the resulting set is [tex]\(\{7, 9, 11, 13, 15\}\)[/tex].
