1. What are Injective (One to One) Functions?

An injective function (also called one-to-one function) is a function where every element of the domain (Set A) is mapped to a distinct element in the codomain (Set B). This means that if f(a) = f(b), then a must equal b.

2. How Does the Calculator Work?

The calculator uses the formula for injective functions:

Where:

• \( n(A) \) — Number of elements in Set A
• \( n(B) \) — Number of elements in Set B
• \( ! \) — Factorial operation

Explanation: The formula calculates the number of ways to assign each element of Set A to a distinct element of Set B without repetition.

3. Importance of Injective Functions

Details: Injective functions are fundamental in mathematics and computer science, particularly in combinatorics, set theory, and database design where unique mappings are required.

4. Using the Calculator

Tips: Enter the number of elements in Set A and Set B. Note that n(B) must be greater than or equal to n(A) for injective functions to exist.

5. Frequently Asked Questions (FAQ)

Q1: What happens if n(B) < n(A)?
A: No injective functions exist from Set A to Set B if n(B) < n(A) because there aren't enough distinct elements in B to map each element of A uniquely.

Q2: How is this different from bijective functions?
A: Injective functions require distinct mappings, while bijective functions are both injective and surjective (onto), meaning they establish a perfect one-to-one correspondence between sets.

Q3: What are some real-world applications?
A: Injective functions are used in encryption, database key constraints, assignment problems, and any situation requiring unique identifiers.

Q4: Can injective functions have empty sets?
A: Yes, there is exactly one function from the empty set to any set (including itself), and it is vacuously injective.

Q5: How does this relate to permutations?
A: When n(A) = n(B), the number of injective functions equals the number of permutations of Set B, which is n(B)!.