1. Stating the problem: We have relations $R$ and $S$ on the set $\{1,2,3\}$: $$R=\{(1,2),(1,3),(2,3),(3,1),(3,3)\}$$ $$S=\{(1,1),(1,3),(2,1),(2,3),(3,2)\}$$ We want to find $T = ((S \circ \overline{R}))^{-1}$ where $\overline{R}$ is the complement of $R$ on $\{1,2,3\} \times \{1,2,3\}$. 2. Find the complement relation $\overline{R}$ on $\{1,2,3\} \times \{1,2,3\}$: The full set of pairs is all $(x,y)$ with $x,y \in \{1,2,3\}$; there are 9 pairs. Pairs in $R$ are: $(1,2),(1,3),(2,3),(3,1),(3,3)$ So $\overline{R} = \{(x,y) \in \{1,2,3\}^2 \mid (x,y) \notin R\} = \{(1,1),(2,1),(2,2),(3,2),(3,3)\}$ 3. Compute composition $S \circ \overline{R}$: By definition, $(a,c) \in S \circ \overline{R}$ if there exists $b$ such that $(a,b) \in \overline{R}$ and $(b,c) \in S$. Check all $(a,b) \in \overline{R}$ and find corresponding $(b,c) \in S$: - For $(1,1) \in \overline{R}$: look for $(1,c) \in S$ are $(1,1),(1,3)$ so $(1,1),(1,3) \in S \circ \overline{R}$ - For $(2,1) \in \overline{R}$: look for $(1,c) \in S$ are $(1,1),(1,3)$ so $(2,1),(2,3) \in S \circ \overline{R}$ - For $(2,2) \in \overline{R}$: look for $(2,c) \in S$ are $(2,1),(2,3)$ so $(2,1),(2,3) \in S \circ \overline{R}$ - For $(3,2) \in \overline{R}$: look for $(2,c) \in S$ are $(2,1),(2,3)$ so $(3,1),(3,3) \in S \circ \overline{R}$ - For $(3,3) \in \overline{R}$: look for $(3,c) \in S$ are $(3,2)$ so $(3,2) \in S \circ \overline{R}$ Collect all unique pairs in $S \circ \overline{R}$: $$S \circ \overline{R} = \{(1,1),(1,3),(2,1),(2,3),(3,1),(3,2),(3,3)\}$$ 4. Find the inverse $((S \circ \overline{R}))^{-1}$ by swapping pairs: $$T = \{(y,x) \mid (x,y) \in S \circ \overline{R}\} = \{(1,1),(3,1),(1,2),(3,2),(1,3),(2,3),(3,3)\}$$ 5. Write $T$ in Python set notation: ```python T = {(1,1),(3,1),(1,2),(3,2),(1,3),(2,3),(3,3)} ``` This is the final answer.