B'
A = {1, 2, 3, 4, 5, 6, 7}, B = {1, 2, 3, 4, 5}
R1 = {(a, b) ∈ R1 | a > b},
R2 = {(a, b) ∈ R2 | a > =b},
Find R2○R1 and R1○R2

Given the sets \( A = \{1, 2, 3, 4, 5, 6, 7\} \) and \( B = \{1, 2, 3, 4, 5\} \), and the relations \( R_1 \) and \( R_2 \):

- \( R_1 = \{(a, b) \mid a > b\} \)
- \( R_2 = \{(a, b) \mid a \geq b\} \)

The problem asks to find the compositions \( R_2 \circ R_1 \) and \( R_1 \circ R_2 \).

### **1. Composition \( R_2 \circ R_1 \)**
The composition \( R_2 \circ R_1 \) is defined as:

\[
R_2 \circ R_1 = \{(a, c) \mid \exists b \text{ such that } (a, b) \in R_1 \text{ and } (b, c) \in R_2\}
\]

This means for every pair \( (a, c) \) in the composition, there must be an intermediate element \( b \) such that \( a > b \) (from \( R_1 \)) and \( b \geq c \) (from \( R_2 \)).

### **2. Composition \( R_1 \circ R_2 \)**
Similarly, the composition \( R_1 \circ R_2 \) is defined as:

\[
R_1 \circ R_2 = \{(a, c) \mid \exists b \text{ such that } (a, b) \in R_2 \text{ and } (b, c) \in R_1\}
\]

This means for every pair \( (a, c) \) in the composition, there must be an intermediate element \( b \) such that \( a \geq b \) (from \( R_2 \)) and \( b > c \) (from \( R_1 \)).

### **Solution Strategy**
To solve this problem, we need to evaluate each composition by checking the conditions on the elements of the sets \( A \) and \( B \). Specifically:

1. **For \( R_2 \circ R_1 \)**:
   - Find all pairs \( (a, c) \) where \( a > b \) and \( b \geq c \).
   
2. **For \( R_1 \circ R_2 \)**:
   - Find all pairs \( (a, c) \) where \( a \geq b \) and \( b > c \).

I'll calculate both compositions now.The compositions \( R_2 \circ R_1 \) and \( R_1 \circ R_2 \) yield the following sets of pairs:

### **1. Composition \( R_2 \circ R_1 \):**
\[
R_2 \circ R_1 = \{(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (7, 1), (7, 2), (7, 3), (7, 4), (7, 5)\}
\]

### **2. Composition \( R_1 \circ R_2 \):**
\[
R_1 \circ R_2 = \{(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (7, 1), (7, 2), (7, 3), (7, 4)\}
\]

### **Observations:**
- **\( R_2 \circ R_1 \)** contains more pairs, specifically those involving the largest elements in \( A \), where the final element \( c \) can be equal to the intermediate element \( b \).
- **\( R_1 \circ R_2 \)** is missing the pairs \( (6, 5) \) and \( (7, 5) \) because there is no intermediate element \( b \) where \( b > 5 \).

If you'd like further details or have questions, feel free to ask!

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Here are 5 related questions for deeper exploration:
1. How would the compositions change if \( A \) and \( B \) had different elements?
2. What happens if we reverse the conditions for \( R_1 \) and \( R_2 \)?
3. Can you generalize a formula to compute the compositions for any sets \( A \) and \( B \)?
4. How does the concept of composition relate to functions in mathematics?
5. What is the effect of restricting the relations to specific subsets of \( A \) and \( B \)?

**Tip:** Composition of relations can be seen as chaining conditions, where the output of one relation serves as the input to another. This is conceptually similar to function composition.

Understand the composition of relations R1 and R2 with detailed examples. Learn how to find R2○R1 and R1○R2 using sets A and B. Explore mathematical concepts of relations and their compositions.

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