1. **Problem:** Let $R$ be a relation on the set $A = \{1,2,3,4,5,6\}$ defined by $R = \{(a,b) : a + b \leq 9\}$. **i) List the elements of $R$.** 2. The set $A$ contains numbers from 1 to 6. We need to find all pairs $(a,b)$ where $a,b \in A$ and $a+b \leq 9$. 3. We check each pair: - For $a=1$: $b$ can be $1,2,3,4,5,6$ since $1+b \leq 9$ for all $b$ in $A$. - For $a=2$: $b$ can be $1,2,3,4,5,6$ since $2+b \leq 9$ for all $b$ in $A$. - For $a=3$: $b$ can be $1,2,3,4,5,6$ since $3+b \leq 9$ for all $b$ in $A$. - For $a=4$: $b$ can be $1,2,3,4,5$ since $4+6=10 > 9$. - For $a=5$: $b$ can be $1,2,3,4$ since $5+5=10 > 9$. - For $a=6$: $b$ can be $1,2,3$ since $6+4=10 > 9$. 4. So the elements of $R$ are: $$ \{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6), (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,4),(4,5), (5,1),(5,2),(5,3),(5,4), (6,1),(6,2),(6,3)\} $$ **ii) Is $R = R^{-1}$?** 5. The inverse relation $R^{-1} = \{(b,a) : (a,b) \in R\}$. 6. Since $a+b \leq 9$ is symmetric in $a$ and $b$, if $(a,b) \in R$ then $(b,a) \in R$. 7. Therefore, $R = R^{-1}$. **Final answers:** - Elements of $R$ are as listed above. - $R$ is equal to its inverse $R^{-1}$.