If `A={2,3,4,5,B={1,3,5,7}` and a relation `R:AtoB` such that `y=2x-3,xepsilonA,yepsilonB`, then find `R`.

Explanation

To find the relation \( R \) from set \( A \) to set \( B \) defined by the equation \( y = 2x - 3 \), we will follow these steps:

### Step 1: Identify the sets
Given:
- Set \( A = \{2, 3, 4, 5\} \)
- Set \( B = \{1, 3, 5, 7\} \)

### Step 2: Define the relation
The relation \( R \) is defined such that \( R = \{(x, y) | y = 2x - 3, x \in A, y \in B\} \).

### Step 3: Calculate \( y \) for each \( x \) in \( A \)
We will substitute each element of \( A \) into the equation \( y = 2x - 3 \) and check if the resulting \( y \) is in set \( B \).

1. For \( x = 2 \):
\[ y = 2(2) - 3 = 4 - 3 = 1 \]
Since \( 1 \in B \), we have the pair \( (2, 1) \).

2. For \( x = 3 \):
\[ y = 2(3) - 3 = 6 - 3 = 3 \]
Since \( 3 \in B \), we have the pair \( (3, 3) \).

3. For \( x = 4 \):
\[ y = 2(4) - 3 = 8 - 3 = 5 \]
Since \( 5 \in B \), we have the pair \( (4, 5) \).

4. For \( x = 5 \):
\[ y = 2(5) - 3 = 10 - 3 = 7 \]
Since \( 7 \in B \), we have the pair \( (5, 7) \).

### Step 4: Compile the relation
Now we can compile all the pairs we found into the relation \( R \):
\[ R = \{(2, 1), (3, 3), (4, 5), (5, 7)\} \]

### Final Answer
The relation \( R \) is:
\[ R = \{(2, 1), (3, 3), (4, 5), (5, 7)\} \]

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