21.1.13. Injective Mapping Proof

Try these first

1. 1.

   Define Catalan numbers.

   Hint

   Refer to their applications in counting.

2. 2.

   What is an injective mapping?

   Hint

   Think about the uniqueness of outputs.

3. 3.

   What is the first Catalan number?

   • 1
   • 2
   • 0
   • 5

   Hint

   Think of the simplest valid pairing.

4. 4.

   True or False: An injective mapping can map two distinct elements to the same element.

   • True
   • False

   Hint

   Recall the definition of injective functions.

5. 5.

   Prove that the reflection method applies to sequences of parentheses and yields a unique transformation.

   Hint

   Reflect on the rules governing the critical points in the sequences.

6. 6.

   Apply the concepts learned to derive the Catalan number for n = 3 and explain your steps.

   Hint

   Utilize both the binomial coefficients and their relationships.