If A={a,b,c,d},B=(p,q,r,s), then which of the following are relations from A to B ? Give reasons for your answer :

Options

A. `R_(1)={(a,p),(b,r),(c,s)}`
B. `R_(2)={q,b),(c,s),(d,r)}`
C. `R_(3)={(a,p),(a,q),(d,p),(c,r),(b,r)}`
D. `R_(4)={(a,p),(q,a),(b,s),(s,b)}`

Answer

To determine which of the given relations are valid from set A to set B, we need to check if each relation consists of ordered pairs where the first element comes from set A and the second element comes from set B.

Given:
- Set A = {a, b, c, d}
- Set B = {p, q, r, s}

Now, let's analyze each relation one by one.

### Step 1: Check Relation R1
**R1 = {(A, P), (B, R), (C, S)}**

- (A, P): A is from A, P is from B. Valid.
- (B, R): B is from A, R is from B. Valid.
- (C, S): C is from A, S is from B. Valid.

**Conclusion for R1:** All pairs are valid. Therefore, R1 is a relation from A to B.

### Step 2: Check Relation R2
**R2 = {(Q, B), (C, S), (D, R)}**

- (Q, B): Q is NOT from A (Q is from B). Invalid.
- (C, S): C is from A, S is from B. Valid.
- (D, R): D is from A, R is from B. Valid.

**Conclusion for R2:** Since the first pair is invalid, R2 is NOT a relation from A to B.

### Step 3: Check Relation R3
**R3 = {(A, P), (A, Q), (D, P), (C, R), (B, R)}**

- (A, P): A is from A, P is from B. Valid.
- (A, Q): A is from A, Q is from B. Valid.
- (D, P): D is from A, P is from B. Valid.
- (C, R): C is from A, R is from B. Valid.
- (B, R): B is from A, R is from B. Valid.

**Conclusion for R3:** All pairs are valid. Therefore, R3 is a relation from A to B.

### Step 4: Check Relation R4
**R4 = {(A, P), (Q, A), (B, S)}**

- (A, P): A is from A, P is from B. Valid.
- (Q, A): Q is NOT from A (Q is from B). Invalid.
- (B, S): B is from A, S is from B. Valid.

**Conclusion for R4:** Since the second pair is invalid, R4 is NOT a relation from A to B.

### Final Conclusion
The valid relations from A to B are:
- R1
- R3