To identify which set contains the perfect square integers between 1 and 80 inclusive, we need to determine which numbers are indeed perfect squares within the provided range. A perfect square is an integer that is the square of another integer.
Let's evaluate the sets provided to see which one contains all such perfect squares:
1. The first set is [tex]\(\{1, 4, 9, 16, 25, 36, 49, 64, 80\}\)[/tex]:
- [tex]\(1 = 1^2\)[/tex]
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
- [tex]\(16 = 4^2\)[/tex]
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(36 = 6^2\)[/tex]
- [tex]\(49 = 7^2\)[/tex]
- [tex]\(64 = 8^2\)[/tex]
- [tex]\(80\)[/tex] is not a perfect square because [tex]\( \sqrt{80} \approx 8.94 \)[/tex] which is not an integer.
2. The second set is [tex]\(\{1, 4, 9, 16, 25, 36, 49, 64\}\)[/tex]:
- [tex]\(1 = 1^2\)[/tex]
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
- [tex]\(16 = 4^2\)[/tex]
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(36 = 6^2\)[/tex]
- [tex]\(49 = 7^2\)[/tex]
- [tex]\(64 = 8^2\)[/tex]
3. The third set is [tex]\(\{4, 9, 16, 25, 36, 49, 64\}\)[/tex]:
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
- [tex]\(16 = 4^2\)[/tex]
- [tex]\(25 = 5^2\)[/tex]
- [tex]\(36 = 6^2\)[/tex]
- [tex]\(49 = 7^2\)[/tex]
- [tex]\(64 = 8^2\)[/tex]
- This set is missing [tex]\(1\)[/tex], which is also a perfect square.
After examining the elements, it is clear tha the set [tex]\(\{1, 4, 9, 16, 25, 36, 49, 64\}\)[/tex] correctly lists all the perfect squares up to 80 and does not include any non-perfect squares like 80.
Thus, the correct set of perfect square integers between 1 and 80 inclusive is:
[tex]\[
\{1, 4, 9, 16, 25, 36, 49, 64\}
\][/tex]
