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 after R"), we will follow these steps:

### Step 1: Identify the Relations
We have two relations:
- \( R = \{(1, 3), (2, 2), (3, 2)\} \)
- \( S = \{(2, 1), (3, 2), (2, 3)\} \)

### Step 2: Write the Mappings for Each Relation
From relation \( R \):
- \( R(1) = 3 \)
- \( R(2) = 2 \)
- \( R(3) = 2 \)

From relation \( S \):
- \( S(2) = 1 \)
- \( S(3) = 2 \)
- \( S(2) = 3 \) (Note: This is a duplicate for \( S(2) \))

### Step 3: Compute \( S \circ R \)
To compute \( S \circ R \), we need to find \( S(R(x)) \) for each \( x \) in the set \( A = \{1, 2, 3\} \).

1. For \( x = 1 \):
- \( R(1) = 3 \)
- Now find \( S(3) \):
- \( S(3) = 2 \)
- Therefore, \( S(R(1)) = S(3) = 2 \).

2. For \( x = 2 \):
- \( R(2) = 2 \)
- Now find \( S(2) \):
- \( S(2) = 1 \) (we can take the first occurrence)
- Therefore, \( S(R(2)) = S(2) = 1 \).

3. For \( x = 3 \):
- \( R(3) = 2 \)
- Now find \( S(2) \):
- \( S(2) = 1 \) (again, we take the first occurrence)
- Therefore, \( S(R(3)) = S(2) = 1 \).

### Step 4: Compile the Results
Now we can compile the results of \( S \circ R \):
- \( S \circ R(1) = 2 \)
- \( S \circ R(2) = 1 \)
- \( S \circ R(3) = 1 \)

Thus, the relation \( S \circ R \) can be expressed as:
\[ S \circ R = \{(1, 2), (2, 1), (3, 1)\} \]

### Final Answer
The composition \( S \circ R \) is:
\[ S \circ R = \{(1, 2), (2, 1), (3, 1)\} \]