To determine whether [tex]\( R \subset A \)[/tex] (i.e., [tex]\( R \)[/tex] is a subset of [tex]\( A \)[/tex]), we need to check if every element in set [tex]\( R \)[/tex] is also an element in set [tex]\( A \)[/tex].
Given sets:
1. [tex]\( A = \{ x \mid x \text{ is an odd integer} \} \)[/tex]
2. [tex]\( R = \{3, 7, 11, 27\} \)[/tex]
Step-by-Step Solution:
1. List the elements of set [tex]\( R \)[/tex]:
[tex]\[
R = \{3, 7, 11, 27\}
\][/tex]
2. Recognize the definition of set [tex]\( A \)[/tex]:
[tex]\[
A = \{ x \mid x \text{ is an odd integer} \}
\][/tex]
This means [tex]\( A \)[/tex] contains all odd integers.
3. Determine if each element in [tex]\( R \)[/tex] is in [tex]\( A \)[/tex]:
- [tex]\( 3 \)[/tex]: This number is an odd integer and is therefore in [tex]\( A \)[/tex].
- [tex]\( 7 \)[/tex]: This number is an odd integer and is therefore in [tex]\( A \)[/tex].
- [tex]\( 11 \)[/tex]: This number is an odd integer and is therefore in [tex]\( A \)[/tex].
- [tex]\( 27 \)[/tex]: This number is an odd integer and is therefore in [tex]\( A \)[/tex].
Since all the elements in [tex]\( R \)[/tex] (i.e., 3, 7, 11, 27) are indeed odd integers, they all belong to set [tex]\( A \)[/tex].
Thus, every element of [tex]\( R \)[/tex] is contained in [tex]\( A \)[/tex], meaning [tex]\( R \)[/tex] is a subset of [tex]\( A \)[/tex], or [tex]\( R \subset A \)[/tex].
The correct answer is:
[tex]\[
\text{yes, because all the elements of set } R \text{ are in set } A
\][/tex]
