MATH 240: Discrete structures I. Fall 2011
Assignment #1: Sets and Logic. Solutions.
1.
Set Identities. Prove the following set identities.
a) (A B) C = (A C) (B C),
b) (A B) (A C) = B C.
Solution a): Consider x (A B) C. We have x A B and x C.
Therefore,
5|@|i4i?| @?_ 6ht| hLLu Lu
hhL<t A iLhi4
Ohw D @ iD> E> => Fj eh d qlwh vhw ri dw ohdvw wkuhh dowhuqdwlyhv1 D wudqvlwlyh
suhihuhqfh ryhu D lv d udqnlqj ri wkh dowhuqdwlyhv lq D iurp wrs wr erwwrp/ zlwk wlhv
doorzhg1 Zh frqvlghu d vrflhw| zlwk Q lqglylgxd
Notes on Logic
1 Propositional Calculus
A proposition or statement is an assertion which can be determined to be either true or false (T or F). For example, zero is less than any positive number is a statement. We are interested in combining and simplifyi
MATH 240: Discrete structures I. Fall 2011
Assignment #4: Cryptography and combinatorics. Solutions.
1.
Fermat primality test.
A number m passes the Fermat primality test if 2m1 1(mod m).
a) Does m = 2047 pass the test?
b) Did the test give the correct an
MATH 240: Discrete structures I. Fall 2011
Assignment #2: Proofs. Due Friday, October 12th.
1.
a)
Problems in NP. Show that the following problems are in NP.
The Knapsack problem.
Given n items numbered 1 through n so that i-th item has weight wi and valu
MATH 240: Discrete structures I. Fall 2011
Assignment #3: Number theory. Solutions.
3
1.
Prime factorization. Show that
2.
Euclids algorithm. Use the Euclids Algorithm to nd each of the following.
3 is irrational.
Solution: Suppose for a contradiction tha
MATH 240: Discrete structures I. Fall 2011
Assignment #2: Proofs. Solutions.
1.
Problems in NP. Show that the following problems are in NP.
a) The Knapsack problem.
Given n items numbered 1 through n so that i-th item has weight wi and value vi ,
determin
MATH 240: Discrete structures I. Fall 2011
Assignment #5: Combinatorics and Graph Theory. Solutions.
1.
Fibonacci Numbers.
Show that for every positive integer n the Fibonacci number F5n is divisible by 5.
Solution: By induction on n. Base case (n = 1): F
MATH 240: Discrete structures I. Fall 2011
Assignment #5: Combinatorics and Graph Theory.
Due Monday, November 28th.
1.
Fibonacci Numbers.
Show that for every positive integer n the Fibonacci number F5n is divisible by 5.
2.
Recurrence relations.
(a) Solv
Further Notes on RSA Cryptosystem
Bob chooses a public lock as follows. He picks 3 large distinct primes p, q1 , q2
with p > q1 q2 . (E.g., think of q1 , q2 as 2-digit primes and p as a 400 digit
prime. We saw how to use our randomized algorithm for prime
Math 240: Discrete Structures I
Wednesday 17th October
Midterm Exam
Instructions. The exam is 50 minutes long and contains 2 questions. Write your answers clearly in the notebook provided. You may quote any result/theorem seen in the lectures or in the as