If `A={a,b,c,d},B={1,2,3}`, find whether or not the following sets of orderd pairs are relations from A to B or not
`R_(2)={(b,1),(c,2),(d,1)}`

Answer

To determine whether the set of ordered pairs \( R_2 = \{(b, 1), (c, 2), (d, 1)\} \) is a relation from set \( A \) to set \( B \), we need to follow these steps:

### Step 1: Define the Sets
We have the following sets:
- \( A = \{a, b, c, d\} \)
- \( B = \{1, 2, 3\} \)

### Step 2: Find the Cartesian Product \( A \times B \)
The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set \( A \) and the second element is from set \( B \).

Calculating \( A \times B \):
\[ A \times B = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3), (d, 1), (d, 2), (d, 3)\} \]

### Step 3: Check if \( R_2 \) is a Subset of \( A \times B \)
Now we need to check if every ordered pair in \( R_2 \) is also in \( A \times B \).

The ordered pairs in \( R_2 \) are:
- \( (b, 1) \)
- \( (c, 2) \)
- \( (d, 1) \)

We will check each pair:
- \( (b, 1) \) is in \( A \times B \)
- \( (c, 2) \) is in \( A \times B \)
- \( (d, 1) \) is in \( A \times B \)

Since all pairs in \( R_2 \) are found in \( A \times B \), we conclude that:
\[ R_2 \subseteq A \times B \]

### Step 4: Conclusion
Since \( R_2 \) is a subset of \( A \times B \), we can say that \( R_2 \) is indeed a relation from set \( A \) to set \( B \).

### Final Answer
Yes, \( R_2 \) is a relation from \( A \) to \( B \).

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