Due at the prelim on 10/9/03.
With the limitations of ascii, we'll use = to mean 'equivalent' if the equation is
written 'mod n'.
1. Let a, b, and c be positive integers with a and b being coprime.
(i) prove that a|bc => a|c
(ii) prove that a|c
cs 280 hw2
Do question 8 from hw1.
2. Prove that
(1+a)^p = (1 + a^p) mod p
<= a < p.
Is it still true if p is not prime?
if p is prime and a is an integer, 0
3. How many different binary operations (recall the formal definition of a
function) are there
Solutions for CS 280 HW 1
(There are several ways to prove the problems involving set equations.
This is a representative solution, others are most certainly acceptable)
Given A+B = (A-B) U (B A) as defined in the problem, prove that:
A+B = ? A
1) Recall that any permutation P can be expressed as a composition of
disjoint cycles. Since these cycles are disjoint, we will present here
an algorithm to express a single cycle as a composition of
transpositions. Let the cycle be (
To make our earlier (boxed) remark more precise, if is an equivalence relation on A, then
for any a A we dene the set [a] = cfw_ x A | x ~ a to be the equivalence class of a and
the set A/~ = cfw_ [a] | a A to be the quotient of A by the relation.
There are a lot of niceties in dening what a set should be, but for now well leave that to the
more esoteric regions of the foundations of mathematics, and be satised with the inherently
problematic denition that a set is a collection of objects dened by
Consider the set 2 = of ordered pairs of integers. We call this a two-dimensional
integer lattice (think grid lines on graph paper). Now consider the relation on this dened by
(a , b) ~ (c , d) ad bc = 0. Its easy to show that this is an equivalence rela
There are some amusing things we can do to build up a collection of sets from a starting
ingredient. For example, lets start with the set A = , and notice that the set B = cfw_A = cfw_
is different from the set A itself. In particular, the set A has noth