2.4.1. Part (a): Counting Functions

Try these first

1. 1.

   Define an injective function and give an example.

   Hint

   Think of a simple linear function.

2. 2.

   What is a surjective function?

   Hint

   Consider mappings to fewer elements.

3. 3.

   What defines a surjective function?

   Hint

   Think of it as 'onto' mapping.

4. 4.

   If a function has 5 elements in the domain and 3 in the codomain, how many injective functions can it have?

   • 1
   • 0
   • 6

   Hint

   Consider if you can assign unique outputs for all inputs.

5. 5.

   Prove that if you're given a surjective function from a finite set A (with 4 elements) to another finite set B (with 3 elements), A cannot contain an injective function.

   Hint

   Reflect on the definitions of injective versus surjective.

6. 6.

   Calculate the number of surjective functions from set A with 5 elements to set B with 3 elements using Stirling numbers.

   Hint

   Use the Stirling function properties and recall how permutations affect partitions.