1. **Problem statement:** Given the set $A = \{1,2,3,4\}$ and the relation $R = \{(a,b) \mid a < b\}$ on integers, find: a) The inverse relation $R^{-1}$. b) The union $R \cup R^{-1}$. 2. **Recall definitions:** - The inverse relation $R^{-1}$ consists of all pairs $(b,a)$ such that $(a,b) \in R$. - The union $R \cup R^{-1}$ contains all pairs in $R$ or in $R^{-1}$. 3. **Find $R$ on $A$:** Since $R = \{(a,b) \mid a < b\}$ and $a,b \in A$, list all pairs where $a < b$: $$R = \{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\}$$ 4. **Find $R^{-1}$:** By definition, invert each pair: $$R^{-1} = \{(2,1),(3,1),(4,1),(3,2),(4,2),(4,3)\}$$ 5. **Find $R \cup R^{-1}$:** Combine all pairs from $R$ and $R^{-1}$: $$R \cup R^{-1} = \{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(2,1),(3,1),(4,1),(3,2),(4,2),(4,3)\}$$ **Final answers:** - $R^{-1} = \{(2,1),(3,1),(4,1),(3,2),(4,2),(4,3)\}$ - $R \cup R^{-1} = \{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(2,1),(3,1),(4,1),(3,2),(4,2),(4,3)\}$