Download Math 232 Projects

Math 232 Projects
1. Irrationality of the number e .
a.
b.
c.
d.
Prove that the number e is irrational.
Deduce from part (a) that ne is irrational for every integer n .
Deduce from part (a) that re is irrational for every rational number r .
Is e 2 irrational.
2. Fibonacci Sequence.
a. State the Rabbit Problem, solve it, and then prove your solution.
b. The Fibonacci sequence is the sequence defined recursively by
F1  1, F2  1, and Fn  Fn1  Fn2 for n  3. Prove the following about
this sequence. In what follows,   (1  5 ) / 2 and   (1  5 ) / 2 .
i. Prove that Fn 
n
ii. Prove that
F
k 1
k
 n  n
5
for n  1.
 Fn  2  1 for all n  1.
iii. Prove that F3n is even for all n  1.
iv. Fn 1 Fn 1  Fn2  (1) n for all n  2 .
3. Towers of Hanoi.
Consider a board containing three upright pegs (a peg is a cylindrical pin of
wood or metal fitted on, say, a board as a support). The first pig contains n
rings, with different diameters, stacked on each other in a decreasing order of
diameters of the rings; That is, the ring with the largest diameter is at the
bottom and the ring with the smallest diameter is on the top. The Goal is to
transfer the rings, one at a time, to the third peg in such a way that the position
of the rings will be the same as the original position in the first peg. During the
transfer, you must not place a larger ring on a smaller ring (and this is why
you have a second peg.) Let M n be the minimum number of moves required
to make the transfer of n rings. Find M n and prove your answer.
4. The Binomial Coefficients.
a.
b.
c.
d.
Read section 4.2 in your textbook.
State and prove the Binomial Theorem.
Solve problems # 4(b), 5(b), 6(b) in page 157 of your textbook.
Prove that 5 | n 5  n for all n  1.
5. Triangular Numbers. Solve Exercise # 18, p.170 in your Textbook.
6. Transfinite Induction.
Let R be a relation on a nonempty set X .
a. What is a partial order on X . What is a total order on X .
b. Give examples of partial and total orders.
c. What is a partially ordered set. Give an example of a partially ordered
set.
d. Does every partially ordered set have a smallest element.
e. State the definition of a well-ordered set for partially ordered sets.
f. Prove the Principle of Transfinite Induction: Let X be a well-ordered
set, and let P (x ) be a predicate which is either true or false for each
element x  X . If for any x  X , P ( y ) is true for every y  x implies
P (x ) is true, then P (x ) is true for every x  X .
7. Infinitude of Primes. Prove that the number of primes is infinite.
8. Cardinality of Power Sets. Let A be a set consisting of n elements. What is
the number of elements in the power set of A . Prove your answer.
All the best,
Dr. Ibrahim Al-Rasasi
Semester 061.