Let's analyze the sets [tex]\( A \)[/tex] and [tex]\( R \)[/tex] based on their definitions.
1. Definition of Set [tex]\( A \)[/tex]:
[tex]\( A = \{x \mid x \text{ is an odd integer} \} \)[/tex]
This means set [tex]\(A\)[/tex] contains all odd integers. For example: [tex]\(\ldots, -3, -1, 1, 3, 5, 7, 9, \ldots\)[/tex].
2. Definition of Set [tex]\( R \)[/tex]:
[tex]\( R = \{3, 7, 11, 27\} \)[/tex]
This set contains exactly the elements 3, 7, 11, and 27.
3. Subset Definition:
A set [tex]\( R \)[/tex] is a subset of set [tex]\( A \)[/tex], written as [tex]\( R \subset A \)[/tex], if and only if every element in [tex]\( R \)[/tex] is also an element in [tex]\( A \)[/tex].
Let's check each element of [tex]\( R \)[/tex] to see if it belongs to [tex]\( A \)[/tex]:
- The element [tex]\( 3 \)[/tex] is an odd integer.
- The element [tex]\( 7 \)[/tex] is an odd integer.
- The element [tex]\( 11 \)[/tex] is an odd integer.
- The element [tex]\( 27 \)[/tex] is an odd integer.
Since all the elements of [tex]\( R \)[/tex] (i.e., 3, 7, 11, and 27) are indeed elements of [tex]\( A \)[/tex], we can conclude that [tex]\( R \subset A \)[/tex].
Given the interpretations of the options:
- "yes, because all the elements of set [tex]\( A \)[/tex] are in set [tex]\( R \)[/tex] " is incorrect because not all elements of [tex]\( A \)[/tex] need to be in [tex]\( R \)[/tex].
- "yes, because all the elements of set [tex]\( R \)[/tex] are in set [tex]\( A \)[/tex]" is correct because it aligns with our finding.
- "no, because each element in set [tex]\( A \)[/tex] is not represented in set [tex]\( R \)[/tex]" is incorrect because this is not a requirement for [tex]\( R \subset A \)[/tex].
- "no, because each element in set [tex]\( R \)[/tex] is not represented in set [tex]\( A \)[/tex]" is incorrect since we've already verified that all elements of [tex]\( R \)[/tex] are in [tex]\( A \)[/tex].
Therefore, the correct choice is:
Yes, because all the elements of set [tex]\( R \)[/tex] are in set [tex]\( A \)[/tex].
