### Problem 7 Suppose \( R \) is the relation on \( \mathbb{Z} \times \mathbb{Z} \) where \( aRb \) means that \( a \) has the same lowest order (right-most) digit as \( b \). **Example:** \((a, b) \in R\) when \( a = 1234 \) and \( b = 64 \) because both numbers end in 4. Determine whether \( R \) is an equivalence relation. Show your reasoning. --- ### Solution Yes, \( R \) is an equivalence relation. - **Reflexive:** \( a \) ends with the same digit as itself, so \((a, a) \in R\). - **Symmetric:** If \( a \) ends with the same digit as \( b \), then \( b \) ends with the same digit as \( a \). Thus, if \((a, b) \in R\), then \((b, a) \in R\). - **Transitive:** If \( a \) ends with the same digit as \( b \), and \( b \) ends with the same digit as \( c \), then \( a \) ends with the same digit as \( c \). Hence, \((a, c) \in R\). If asked this on the final, you must show proof of all three required properties.