If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

Options

A. 720
B. 120
C. 0
D. None of these

Answer

To solve the problem of finding the number of one-one and onto mappings from set A to set B, we will follow these steps:

### Step 1: Understand the Definitions
- A **one-one (injective)** mapping means that each element of set A maps to a unique element in set B. No two elements in A can map to the same element in B.
- An **onto (surjective)** mapping means that every element in set B must be mapped by at least one element in set A.

### Step 2: Analyze the Sets
- Set A has 5 elements: \( |A| = 5 \)
- Set B has 6 elements: \( |B| = 6 \)

### Step 3: Determine the Possibility of One-One and Onto Mappings
For a mapping from set A to set B to be both one-one and onto:
- The number of elements in set A must be equal to the number of elements in set B. This is because:
- For a one-one mapping, each element in A must map to a unique element in B.
- For an onto mapping, every element in B must be mapped from A.

Since \( |A| = 5 \) and \( |B| = 6 \), it is impossible to have a mapping that is both one-one and onto because there are not enough elements in set A to cover all elements in set B.

### Step 4: Conclusion
Thus, the number of one-one and onto mappings from A to B is **0**.

### Final Answer
The number of one-one and onto mappings from set A to set B is **0**.

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