21.1.10. Constructing Sequence S'

Try these first

1. 1.

   Define a bad sequence in the context of sequence S.

   Hint

   Think of sequences that do not meet the criteria of non-negative sums.

2. 2.

   What is the cardinality of set A?

   Hint

   Remember this set includes all combinations of the 1s and -1s.

3. 3.

   What describes a bad sequence?

   • Has no negative sums
   • Has at least one occurrence of a negative sum
   • Is a sequence of only 1s

   Hint

   Think about what makes a sequence invalid.

4. 4.

   True or False: The reflection method creates an injective mapping.

   • True
   • False

   Hint

   Reflect on the one-to-one relationship established by this process.

5. 5.

   Design a sequence of length 6 involving three 1s and three -1s. Show it's a bad sequence and apply the reflection method to construct S'. What results do you observe in the reconstruction?

   Hint

   Make sure to track where your sums dip low.

6. 6.

   Apply subtractive counting to establish a scenario problem with n=5. Create conditions to define valid versus invalid sequences, and derive overall conclusions from counts.

   Hint

   Think of it like evaluating checks and balances within defined counting principles.