## MATH 6: Finite Mathematics Last updated August 19, 2007

Homework and Handouts

Remember: show your work and explain your answers! If you have questions about what needs to be said, do not hesitate to contact me.

Due Monday, July 2:
1.1(b): 1, 2
1.4(b): 5, 7, 10(i)-(v), 12
1.5(b): 1, 6, 9, 12, 13
2.1(b): 5, 8, 11, 12, 16
2.2(b): 2, 3, 5, 8

Due Monday, July 9:
2.3(b): 1, 2, 5, 10
2.4(b): 1, 3, 6, 8

Due Tuesday, July 17 (in my mailbox by 5 PM):
3.1(b): 4(i) and (ii), 6
3.2(b): 1, 6, 7
3.3(b): 1, 2, 4, 9, 12, 14
3.4(b): 1, 5, 6

Also do the following extra problem for (3.3)(b): Six students decide to play the game of ``you're the oddball.'' In this game each person flips a (fair, two-sided) coin. If there is someone whose flip does not match anyone else's, he is called the oddball and is thrown out. If no one is thrown out, then the game is played for another round. What is the probability that the first person to be thrown out is thrown out after the third round?

Due Monday, July 30:
4.3(b): 1, 3, 4
4.4(b): 1, 2, 4

Additional problems (some taken from Matters Mathematical by Herstein and Kaplansky):

Here R denotes the set of real numbers.

1. Suppose S = T = the set of all people alive on planet Earth today. a) Which of the following define functions from S to T?
1. f(s) = father of s,
2. f(s) = oldest living sister of s,
3. f(s) = oldest living member of the immediate family of s.
b) What are the answers to these questions if we keep S the same but replace T with the set of people who ever lived on planet Earth?
c) If f(s) = father of s, how could you describe (f ο f)(s) in words?
2. Consider the function f: RR given by f(x) = x2. a) Is f(x) onto? Is f(x) one-to-one? b) What are the answers to these questions if we replace R (in both the domain and range) with the set of positive real numbers?
3. How many surjective functions are there from a 12-element set to a 2-element set?
4. Suppose that n is at least m. How many injective functions are there from an m-element set to an n-element set?
5. Suppose f is an onto function from S to T. And suppose g is an onto function from the set T to the set U. Is the composition g ο f an onto function from the set S to the set U? If so, give a convincing explanation of why this is the case. If not, give a counterexample.

Extra credit (5 pt): Write down everything you remember about the class Sarah gave (on Hamiltonian cycles and the four-color theorem). What was your favorite thing about the class? Your least favorite? Did you feel the atmosphere was conducive to your participation in class discussions?

Due Monday, August 6:
Problems 1, 2, 3, 4, 5, 9 from the permutations worksheet.

Due Wednesday, August 15:
The HW problems from the modular arithmetic worksheet 1 and from worksheet 2

Due Wednesday, August 22:
The HW problems from the final number theory worksheet. Also Exercises 6, 7 on p. 107 and Exercise 1 on page 114 (from the textbook).

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