Which of the following relation is a function ?

Options

A. {(a,b) (b,e)(c,e) (b,x)}
B. {(a,d)(a,b)(b,e)(a,b)}
C. {(a,d)(b,e)(c,d)(e,x)}
D. {(a,b)(b,m)(b,y)(d,x)}

Answer

To determine which of the given relations is a function, we need to analyze each option based on the definition of a function. A relation is a function if:

1. Every element in set A (domain) is associated with exactly one element in set B (codomain).
2. No two different elements in set A map to the same element in set B.

Let's analyze the options step by step:

### Step 1: Analyze Option A
- **Relation**: {(A, B), (B, E), (C, E), (B, X)}
- **Elements in Set A**: {A, B, C}
- **Elements in Set B**: {B, E, X}

In this relation:
- A maps to B
- B maps to E
- C maps to E
- B maps to X

Here, B is mapped to both E and X, which violates the rule that each element in A must map to a unique element in B. Therefore, **Option A is not a function**.

### Step 2: Analyze Option B
- **Relation**: {(A, D), (A, B), (B, E)}
- **Elements in Set A**: {A, B}
- **Elements in Set B**: {D, B, E}

In this relation:
- A maps to D
- A maps to B
- B maps to E

Again, A is mapped to both D and B, which violates the uniqueness condition. Therefore, **Option B is not a function**.

### Step 3: Analyze Option C
- **Relation**: {(A, D), (B, E), (C, D), (E, X)}
- **Elements in Set A**: {A, B, C, E}
- **Elements in Set B**: {D, E, X}

In this relation:
- A maps to D
- B maps to E
- C maps to D
- E maps to X

Here, A and C both map to D, but since they are different elements in A, it does not violate the uniqueness condition. Each element in A is mapped to an element in B, and no two elements in A map to the same element in B. Therefore, **Option C is a function**.

### Step 4: Analyze Option D
- **Relation**: {(A, B), (B, M), (B, N), (D, X)}
- **Elements in Set A**: {A, B, D}
- **Elements in Set B**: {B, M, N, X}

In this relation:
- A maps to B
- B maps to M
- B maps to N
- D maps to X

Here, B is mapped to both M and N, which violates the uniqueness condition. Therefore, **Option D is not a function**.

### Conclusion
The only relation that satisfies the conditions of being a function is **Option C**.