Homework 7, Due Tuesday, November 7.

    

  1.  Write a definition for the following sequences: (half credit for
   correct recursive definitions).  You do not need to do anything
   with the explanations given.

(a) 64, 60, 56, 52, 48, 44, 40 ...
    (the number of different possible 2 card hands that give blackjack
    when there are zero, one, two, etc... of the 10 valued cards
    removed from the deck)

(b) 136,120,105,91,78,66...  
    (the total number of different possible 2 card hands that include two
    10 valued cards when there are zero, one, two, etc... of the 10
    valued cards removed from the deck)

(c) 5,10,20,40,80,160 (The maximum amount of money you can lose
    playing the martingale "double-your-bet-when-you-lose" strategy
    for one, two, three etc... hands)

(d) 0,2,8,26,80,242,728,2186,6560,19682


  2. If f is a one to one function and f o g is a one to one function,
must g also be a one to one function?  Prove your answer.


  3. let f: R+ ----> R be defined by f(x) = log(x).  Prove
that f is 1-1.  (R+ is the set of real numbers > 0).


  4.  Let A be the set {4,5,6}, and B be the set {1,2,3,4,5}.  Let
P(A) be the "powerset" of A, and P(B) be the powerset of B.

      

    1. How many elements does the set A x B have?

    2. How many elements does A x P(A) have? 

    3. list two different elements of A x P(A).

    4. How many elements does P(A) x P(B) have?