Review 3
Math 2310
Name:
Number:
1. Give an example of a function f : Z → Z both injective and surjective but different from the identity
map.
2. Give an example of a function f : Z → Z that is surjective but not injective.
3. How many non injective functions are there from A = {1, 2, 3, 4} to itself?
4. Use induction to show that a − b is a factor of an − bn for any n > 0.
5. For which positive integers n is n + 6 < n2 − 8n? Prove your conjecture using induction.
6. Prove that n2 − 1 is divisible by 8 whenever n is an odd positive number.
7. The internal telephone number system on a university campus consist of 5 digits such that none of
the first two is either a 0 or a 1. How many phone numbers can be assigned in such a system?
8. How many license plate consisting of 3 letters and 3 digits contains no letter twice and at least two
times the same digit?
9. A hatcheck person decides to hand back hats to 10 persons giving the correct hat only to 3 persons.
In how many ways can he do it?
10. Solve the recurrence relation an = 2an−1 + 1 with initial condition a1 = 1.
11. Solve the recurrence relation an = nan−1 − 1 with initial condition a0 = 1.
12. Solve the recurrence relation an = an−1 + 6an−2 with initial conditions a0 = 3, a1 = 6.
13. Suppose that every hour there are two new bacteria in a colony for each bacterium that was present
the previous hour, and that all bacteria two hours old die. the colony starts with 100 new bacteria.
Set up a recurrence relation for the number of bacteria present after n hours. Which are the initial
conditions? Solve the recurrence relation.