Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two relations on set A = {(1, 2, 3)}. Then, SoR is equal

Options

A. {(2, 3), (3, 2), (2, 1)}
B. {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}
C. {(3, 2), (1, 3)}
D. {(2, 3), (3, 2)}

Explanation

To find the composition of the relations \( S \circ R \) (read as "S of R"), we need to follow these steps:

### Step 1: Identify the relations R and S
Given:
- \( R = \{(1, 3), (2, 2), (3, 2)\} \)
- \( S = \{(2, 1), (3, 2), (2, 3)\} \)

### Step 2: Determine the output of R for each input
We will find the output of relation \( R \) for each element in its domain.

- For \( 1 \): \( R(1) = 3 \)
- For \( 2 \): \( R(2) = 2 \)
- For \( 3 \): \( R(3) = 2 \)

### Step 3: Find the corresponding outputs in S for the outputs of R
Now, we will use the outputs from \( R \) to find the corresponding values in \( S \).

1. For \( R(1) = 3 \):
- We need to find \( S(3) \). From \( S \), we see that \( S(3) = 2 \).

2. For \( R(2) = 2 \):
- We need to find \( S(2) \). From \( S \), we see that \( S(2) = 1 \) and \( S(2) = 3 \). So, we have two outputs: \( 1 \) and \( 3 \).

3. For \( R(3) = 2 \):
- Again, we need to find \( S(2) \). As found earlier, \( S(2) = 1 \) and \( S(2) = 3 \). So, we again have two outputs: \( 1 \) and \( 3 \).

### Step 4: Compile the results to form the relation \( S \circ R \)
From the outputs we found:
- For \( 1 \): \( S(R(1)) = S(3) = 2 \)
- For \( 2 \): \( S(R(2)) = S(2) = 1 \) and \( S(2) = 3 \)
- For \( 3 \): \( S(R(3)) = S(2) = 1 \) and \( S(2) = 3 \)

Thus, the relation \( S \circ R \) can be compiled as:
- From \( 1 \) we get \( (1, 2) \)
- From \( 2 \) we get \( (2, 1) \) and \( (2, 3) \)
- From \( 3 \) we get \( (3, 1) \) and \( (3, 3) \)

### Final Result
Combining all these, we have:
\[ S \circ R = \{(1, 2), (2, 1), (2, 3), (3, 1), (3, 3)\} \]