Let A={1,2,3,4,5} and B= {1,3,5,7,9} which of the following is are relation from A to B ?

Options

A. `R_(1)` = {(a, b) | b = 2 + a, `a inA, b in B`}
B. `R_(2)` ={(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}
C. `R_(3)`={(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}
D. `R_(4)` = {(1,3), (2, 5), (2, 4), (7, 9)}

Answer

To determine which of the given options are relations from set A to set B, we need to understand the definition of a relation. A relation from set A to set B is a subset of the Cartesian product A × B, which consists of ordered pairs (a, b) where a ∈ A and b ∈ B.

Given:
- A = {1, 2, 3, 4, 5}
- B = {1, 3, 5, 7, 9}

Now, let's analyze each option step by step.

### Step 1: Analyze Option A
Let’s denote the relation R1 as described in the problem. It states that for each element \( a \) in A, the corresponding \( b \) in B is given by \( b = 2 + a \).

- For \( a = 1 \): \( b = 2 + 1 = 3 \) (in B)
- For \( a = 2 \): \( b = 2 + 2 = 4 \) (not in B)
- For \( a = 3 \): \( b = 2 + 3 = 5 \) (in B)
- For \( a = 4 \): \( b = 2 + 4 = 6 \) (not in B)
- For \( a = 5 \): \( b = 2 + 5 = 7 \) (in B)

Since 4 and 6 are not in set B, R1 is not a valid relation from A to B.

### Step 2: Analyze Option B
Let’s denote the relation R2 as described in the problem. The relation consists of pairs where the first element is from A and the second element is from B.

- The pairs are: (1, 1), (1, 3), (1, 5), (2, 1), (2, 3), (2, 5), (3, 1), (3, 3), (3, 5), (4, 1), (4, 3), (4, 5), (5, 1), (5, 3), (5, 5).

All the second elements of these pairs (1, 3, 5) are in B. Therefore, R2 is a valid relation from A to B.

### Step 3: Analyze Option C
Let’s denote the relation R3 as described in the problem. The pairs are (1, 1), (1, 3), (1, 5).

- All the second elements of these pairs (1, 3, 5) are in B. Therefore, R3 is also a valid relation from A to B.

### Step 4: Analyze Option D
Let’s denote the relation R4 as described in the problem. The pairs are (1, 2), (1, 7).

- The element 7 is not in set B. Therefore, R4 is not a valid relation from A to B.

### Conclusion
The valid relations from A to B are:
- R2 (Option B)
- R3 (Option C)

### Final Answer
The relations from A to B are: **Option B and Option C.**

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