Given the sets:
[tex]\[ U=\{x \mid x \ \text{is a real number} \} \][/tex]
[tex]\[ A=\{x \mid x \ \text{is an odd integer} \} \][/tex]
[tex]\[ R=\{x \mid x=3, 7, 11, 27\} \][/tex]
To determine if [tex]\( R \subset A \)[/tex], we need to verify if every element in set [tex]\( R \)[/tex] is also an element in set [tex]\( A \)[/tex].
Set [tex]\( A \)[/tex] includes all odd integers. Let's list a few odd integers to clarify:
[tex]\[ A = \{ \ldots, -3, -1, 1, 3, 5, 7, 9, 11, 13, \ldots \} \][/tex]
Set [tex]\( R \)[/tex] explicitly states its elements:
[tex]\[ R = \{3, 7, 11, 27\} \][/tex]
Now, let's check each element of [tex]\( R \)[/tex]:
- 3 is in [tex]\( A \)[/tex] (3 is an odd integer).
- 7 is in [tex]\( A \)[/tex] (7 is an odd integer).
- 11 is in [tex]\( A \)[/tex] (11 is an odd integer).
- 27 is in [tex]\( A \)[/tex] (27 is an odd integer).
Since all elements of [tex]\( R \)[/tex] are in [tex]\( A \)[/tex], we conclude that [tex]\( R \)[/tex] is indeed a subset of [tex]\( A \)[/tex].
Therefore, the answer is:
[tex]\[ \text{Yes, because all the elements of set } R \text{ are in set } A. \][/tex]
