Let A= {1, 2, 3}and R= {(1, 2), (1, 3), (3, 2)}. Determine
R ◦R^−1, Dom(R ◦R^−1) and Ran(R ◦R^−1).

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 (\( \text{Ran}(R \circ R^{-1}) \)).

---

### 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:
\[
R \circ R^{-1} = \{(1, 1), (1, 3), (3, 1), (3, 3)\}.
\]

---

### 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 \( (a, c) \in R \circ R^{-1} \):
\[
\text{Ran}(R \circ R^{-1}) = \{1, 3\}.
\]

---

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

---

### 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.

Let A= {1, 2, 3} and R= {(1, 2), (1, 3), (3, 2)}. Determine R ◦R^−1, Dom(R ◦R^−1) and Ran(R ◦R^−1).

Explore the computation of R ◦ R^−1, the domain, and range for the relation R = {(1, 2), (1, 3), (3, 2)} on the set A = {1, 2, 3}. Learn step-by-step calculations involving set theory, inverse relations, and composition of relations, suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation