If `A={-1, 2,3}` and `B={0,3,5}` then what is the number of bijections from A to B?

Answer

To find the number of bijections from set \( A \) to set \( B \), we need to follow these steps:

### Step 1: Understand the Sets
We have two sets:
- \( A = \{-1, 2, 3\} \) which has 3 elements.
- \( B = \{0, 3, 5\} \) which also has 3 elements.

### Step 2: Define Bijection
A bijection is a function that is both one-to-one (injective) and onto (surjective). This means that every element in set \( A \) must map to a unique element in set \( B \), and every element in set \( B \) must be mapped by some element in set \( A \).

### Step 3: Count the One-to-One Functions
To find the number of one-to-one functions from \( A \) to \( B \):
- The first element of \( A \) (let's say \(-1\)) can be mapped to any of the 3 elements in \( B \).
- The second element of \( A \) (say \(2\)) can then be mapped to any of the remaining 2 elements in \( B \).
- The last element of \( A \) (say \(3\)) can only be mapped to the last remaining element in \( B \).

Thus, the number of one-to-one functions is calculated as:
\[ 3 \times 2 \times 1 = 6 \]

### Step 4: Check for Onto Functions
Since both sets \( A \) and \( B \) have the same number of elements (3), every one-to-one function from \( A \) to \( B \) is also onto. This means that the range of the function will cover all elements in \( B \).

### Conclusion
Since we have established that there are 6 one-to-one functions and they are also onto, the total number of bijections from \( A \) to \( B \) is:
\[ \text{Total Bijections} = 6 \]

### Final Answer
The number of bijections from \( A \) to \( B \) is \( 6 \).

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