To solve the problem, we need to carefully analyze the composition $R \circ R^{-1}$, as well as its domain ($\text{Dom}(R \circ R^{-1})$) and range ($\text{Ran}(R \circ R^{-1})$).

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Step 1: Understand $R^{-1}$ and $R \circ R^{-1}$

Given $R = \{(1, 2), (1, 3), (3, 2)\}$, the inverse relation $R^{-1}$ is obtained by swapping the components of each pair in $R$: $R^{-1} = \{(2, 1), (3, 1), (2, 3)\}.$

The composition $R \circ R^{-1}$ is defined as: $R \circ R^{-1} = \{(a, c) \mid \exists b \text{ such that } (a, b) \in R \text{ and } (b, c) \in R^{-1}\}.$

Step 2: Compute $R \circ R^{-1}$

To compute $R \circ R^{-1}$, consider all $(a, b) \in R$ and $(b, c) \in R^{-1}$:

1. From $R = \{(1, 2), (1, 3), (3, 2)\}$, the possible $b$-values are $2$ and $3$.
2. From $R^{-1} = \{(2, 1), (3, 1), (2, 3)\}$, the $b$-values match with $a$-values from $R$, resulting in $(a, c)$ pairs.

Match and compute:

• For $(1, 2) \in R$ and $(2, 1) \in R^{-1}$: $(1, 1)$ is in $R \circ R^{-1}$.
• For $(1, 2) \in R$ and $(2, 3) \in R^{-1}$: $(1, 3)$ is in $R \circ R^{-1}$.
• For $(1, 3) \in R$ and $(3, 1) \in R^{-1}$: $(1, 1)$ is in $R \circ R^{-1}$ (already included).
• For $(1, 3) \in R$ and $(3, 1) \in R^{-1}$: $(1, 1)$ is repeated.
• For $(3, 2) \in R$ and $(2, 1) \in R^{-1}$: $(3, 1)$ is in $R \circ R^{-1}$.
• For $(3, 2) \in R$ and $(2, 3) \in R^{-1}$: $(3, 3)$ is in $R \circ R^{-1}$.

Thus: $R \circ R^{-1} = \{(1, 1), (1, 3), (3, 1), (3, 3)\}.$

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Step 3: Domain and Range

• The domain of $R \circ R^{-1}$, $\text{Dom}(R \circ R^{-1})$, is the set of all $a$ such that $(a, c) \in R \circ R^{-1}$: $\text{Dom}(R \circ R^{-1}) = \{1, 3\}.$

• The range of $R \circ R^{-1}$, $\text{Ran}(R \circ R^{-1})$, is the set of all $c$ such that $(a, c) \in R \circ R^{-1}$: $\text{Ran}(R \circ R^{-1}) = \{1, 3\}.$

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Final Answer:

1. $R \circ R^{-1} = \{(1, 1), (1, 3), (3, 1), (3, 3)\}$
2. $\text{Dom}(R \circ R^{-1}) = \{1, 3\}$
3. $\text{Ran}(R \circ R^{-1}) = \{1, 3\}$

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Questions:

1. What is the general formula for $R \circ R^{-1}$ in terms of any set $R$?
2. How do inverse relations affect the domain and range of a relation?
3. Can $R \circ R^{-1}$ ever produce a relation that includes elements not in $A \times A$?
4. How would $R \circ R$ differ from $R \circ R^{-1}$ in this context?
5. Can you identify whether $R \circ R^{-1}$ is reflexive, symmetric, or transitive?

Tip: When composing relations, check for matching elements in both sets to simplify calculations.