1. **Problem statement:** Define the relations $R_1$ and $R_2$ from set $A = \{2,4,6,8\}$ to set $B = \{1,2,3,4\}$. 2. **Relation $R_1$:** $R_1 = \{(x,y) \mid x \in A, y \in B, y \text{ divides } x\}$. This means $y$ divides $x$ if $x \mod y = 0$. 3. **Find $R_1$ pairs:** - For $x=2$: $y$ divides 2 if $y=1$ or $2$. - For $x=4$: $y=1,2,4$ divide 4. - For $x=6$: $y=1,2,3$ divide 6. - For $x=8$: $y=1,2,4$ divide 8. So, $$R_1 = \{(2,1),(2,2),(4,1),(4,2),(4,4),(6,1),(6,2),(6,3),(8,1),(8,2),(8,4)\}$$ 4. **Relation $R_2$:** $R_2 = \{(x,y) \mid x \in A, y \in B, x = 2y\}$. 5. **Find $R_2$ pairs:** Check for each $x$ if there exists $y$ such that $x=2y$ and $y \in B$: - $x=2$: $y=1$ (since $2=2\times1$) - $x=4$: $y=2$ - $x=6$: $y=3$ - $x=8$: $y=4$ So, $$R_2 = \{(2,1),(4,2),(6,3),(8,4)\}$$ **Final answers:** - $R_1 = \{(2,1),(2,2),(4,1),(4,2),(4,4),(6,1),(6,2),(6,3),(8,1),(8,2),(8,4)\}$ - $R_2 = \{(2,1),(4,2),(6,3),(8,4)\}$