To determine if the set \( T \) is a subset of the set \( R \), we need to analyze the definitions of both sets:
1. **Definition of Set \( R \)**:
\[
R = \{ x \in \mathbb{Z} \mid x \text{ is divisible by } 2 \}
\]
This means that \( R \) contains all integers that can be expressed in the form \( 2k \) where \( k \) is an integer. Therefore, \( R \) includes numbers like \( \ldots, -4, -2, 0, 2, 4, 6, 8, \ldots \).
2. **Definition of Set \( T \)**:
\[
T = \{ x \in \mathbb{Z} \mid x \text{ is divisible by } 6 \}
\]
This means that \( T \) contains all integers that can be expressed in the form \( 6m \) where \( m \) is an integer. Therefore, \( T \) includes numbers like \( \ldots, -12, -6, 0, 6, 12, 18, \ldots \).
3. **Check if \( T \subseteq R \)**:
To show that \( T \) is a subset of \( R \), we need to show that every element of \( T \) is also an element of \( R \).
- Let \( x \) be an arbitrary element of \( T \). By the definition of \( T \), we can write:
\[
x = 6m \quad \text{for some } m \in \mathbb{Z}
\]
- Since \( 6m \) can be factored as:
\[
x = 2(3m)
\]
where \( 3m \) is also an integer (since \( m \) is an integer), it follows that \( x \) is divisible by \( 2 \).
- Therefore, \( x \) must also be in \( R \).
4. **Conclusion**:
Since every element \( x \) in \( T \) is also in \( R \), we conclude that:
\[
T \subseteq R
\]
