If `A={a, b, c, d} and B={x, y, z}`, then which one of the following relations from A to B is not a mapping?

Options

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

Answer

To determine which relation from set A to set B is not a mapping, we first need to understand the definitions of the sets and the concept of a mapping (or function).

### Given:
- Set A = {a, b, c, d}
- Set B = {x, y, z}

### Definition of Mapping:
A relation from set A to set B is called a mapping (or function) if every element in set A is related to exactly one element in set B. This means:
- Each element in A must map to one and only one element in B.
- No element in A can map to more than one element in B.

### Steps to Identify the Non-Mapping Relation:

1. **List Possible Relations**: We need to analyze the given relations from A to B. Let's denote the relations as follows:
- Relation 1: a → x
- Relation 2: b → y
- Relation 3: c → z
- Relation 4: d → x
- Relation 5: a → y
- Relation 6: b → z
- Relation 7: c → x
- Relation 8: d → z

2. **Check Each Relation**: For each relation, we need to check if any element from set A maps to more than one element in set B:
- If a → x and a → y, then 'a' is mapping to two different elements in B, which is not allowed.
- Similarly, if b → x and b → z, then 'b' is mapping to two different elements in B.

3. **Identify Non-Mapping Relation**:
- If we find any element in A that maps to two different elements in B, that relation is not a mapping.
- For example, if we have a relation like { (a, x), (a, y) }, this means 'a' is related to both 'x' and 'y', which violates the mapping condition.

4. **Conclusion**: After analyzing the relations, we find that if any element in A is related to more than one element in B, that specific relation is not a mapping.

### Example of Non-Mapping Relation:
If we have a relation defined as:
- a → x
- b → y
- c → z
- d → x
- a → y (this is the crucial part)

Here, 'a' is related to both 'x' and 'y', which means this relation is not a mapping.

### Final Answer:
The relation that is not a mapping is the one where at least one element from set A is related to two different elements in set B.

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