Math 240 Assignment 4 Solutions
Assigned on 4 Nov, 2012. Due on 19 Nov, 2012, before 5PM.
1. (RSA Encryption). Recall that in RSA encryption I publicly reveal a prime p and an integer
n that is the product of 2 primes. Together, the pair (p, n) form my pu
Circuit Complexity
3.1 Boolean Logic
_ Objects: statements p, 1
_ Operators: _;^;:, etc
3.2 Logic Gates
Will insert logic gate diagrams later when I _gure how to insert images.
p q r p _ q (p _ q) _ r
11101
11000
10110
10011
01100
01011
00111
00000
Majori
1 Sets
1.1 De_nition
A set is a collection of distinct objects which are called the elements of the set.
Examples: We use a capital letter for sets.
_ A = fAlice;Bob;Claire;Eveg
_ B = fa; e; i; o; ug = fo; i; e; a; ug
_ N = f1; 2; 3; 4; 5; :g (natural num
Math 240 Assignment 2
Assigned on 27 Sep, 2012. Due on Oct 8, 2012, before 5PM.
1. (Order notation.) For each of the following pairs f : N R, g : N R, indicate whether
f (n) = O(g(n), f (n) = (g(n), or both. In each case, briey justify your answer.
(a)
(b
Math 240 Assignment 3
Assigned on 11 Oct, 2012. Due on Oct 29, 2012, before 5PM.
1. (Irrational numbers). Prove that
3
7 is irrational.
2. (Euclids algorithm). Use Euclids algorithm to nd
(a) gcd(583, 297)
(b) gcd(1208, 252)
(c) gcd(55, 34)
3. (Congruence
Math 240 Assignment 5
Assigned on 19 Nov, 2012. Due on 3 Dec, 2012, before 5PM.
1. (Recurrence relations). Find a recurrence for the number of ways to make n cents change
using only pennies, nickels, and dimes. You should have base cases for n = 0, 1, . .
Math 240 Assignment 1 Solutions
Louigi Addario-Berry
1. (Venn Diagrams.)
(a) Draw the Venn diagram for A (BC).
(b) Using as few symbols as possible, write down the sets corresponding to the shaded regions in the
Venn diagrams in the Figure. For the rst yo