21.1.14. Surjective Mapping Proof

Try these first

1. 1.

   Define a bad sequence in terms of 1s and -1s.

   Hint

   Think about the conditions that need to be met for a sequence to be valid.

2. 2.

   What is the binomial coefficient for selecting 2 from 4?

   Hint

   Recall the formula for binomial coefficients C(n, k) = n! / (k!*(n-k)!), where ! denotes factorial.

3. 3.

   What characterizes a surjective mapping?

   • Every element in the domain is mapped
   • Every element in the codomain is mapped
   • Only some elements in the codomain are mapped

   Hint

   Focus on the relationship between domain and codomain.

4. 4.

   True or False: A 'bad' sequence has no negative partial sums at any position.

   • True
   • False

   Hint

   Go back to the definition of a bad sequence.

5. 5.

   Given n = 3, calculate the exact number of valid sequences and explain your methodology.

   Hint

   Break down into steps—first count all, then reflect negative sequences.

6. 6.

   If you were to find patterns among valid sequences, describe the fundamental properties that emerge when using reflective methods.

   Hint

   Consider how alterations in each step lead to consistent patterns in counts.