1. **State the problem:** We have a set $A = \{1,2,3,4,5,6\}$ and a relation $R$ defined by $R = \{(x,y) : y = x + 1, x,y \in A\}$. We want to find all pairs $(x,y)$ in $R$. 2. **Understand the relation:** The relation pairs each element $x$ in $A$ with $y$ such that $y = x + 1$ and $y$ is also in $A$. 3. **Find all pairs:** We check each $x$ in $A$ and see if $x+1$ is in $A$. - For $x=1$, $y=1+1=2 \in A$, so $(1,2) \in R$. - For $x=2$, $y=2+1=3 \in A$, so $(2,3) \in R$. - For $x=3$, $y=3+1=4 \in A$, so $(3,4) \in R$. - For $x=4$, $y=4+1=5 \in A$, so $(4,5) \in R$. - For $x=5$, $y=5+1=6 \in A$, so $(5,6) \in R$. - For $x=6$, $y=6+1=7 \notin A$, so no pair. 4. **Final answer:** $$R = \{(1,2), (2,3), (3,4), (4,5), (5,6)\}$$ This relation contains all pairs where the second element is exactly one more than the first, and both are in the set $A$.