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_(1)={(a,1),(a,3)}`

Answer

To determine whether the set of ordered pairs \( R_1 = \{(a, 1), (a, 3)\} \) is a relation from set \( A \) to set \( B \), we need to follow these steps:

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

### Step 2: Define 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 \( R_1 \) is a Subset of \( A \times B \)
Now we need to check if the set \( R_1 \) is a subset of \( A \times B \).

Given:
\[ R_1 = \{(a, 1), (a, 3)\} \]

We can see that both elements of \( R_1 \) are present in \( A \times B \):
- \( (a, 1) \) is in \( A \times B \)
- \( (a, 3) \) is in \( A \times B \)

### Step 4: Conclusion
Since all elements of \( R_1 \) are found in \( A \times B \), we conclude that \( R_1 \) is indeed a relation from \( A \) to \( B \).

### Final Answer
Yes, \( R_1 \) is a relation from \( A \) to \( B \).

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