October 5, 1995

		cs330 - Discrete Structures
		===========================
		          Fall 1995


Homework #3
Due: October 16, 1995

Exercise	Page	Points
------------------------------
20		79	20
2		88	30
4		88	 5
14		88	30
20		88	15
Problem 1 	below	10
Problem 2	below	30
------------------------------
		Total  140

1) Let F(N) denote the set of all finite subsets of N (the set of
natural numbers). We want to show that F(N) is countable; to do this
we must find a bijection between between F(N) and N (this is what the
definitions says). Another way to prove F(N) is countable is to find
a bijection between F(N) and a set proven to be countable.
Here we go: let B be the set of binary strings with no leading zeros,
except for number 0. Example of strings in B are:
	  0	the binary representation of 0
	  1	the binary representation of 1
	 01	invalid because it has a leading 0
	 10	the binary representation of 2
	101	the binary representation of 5
Obviously B is countable because each string correspons to a natural
number. We define a function from B tp F(N) as follws:
	g:B -> F(N)
	g(bn...b1b0) = {i | bi=1}
For instance
	g(0) = {}
	g(1) = {0}
	g(10) = {1}
	g(11) = {0, 1}
	g(110) = {1, 2}
Show that g is a bijection.

2) For each of the following functions, state which one of the following
properties holds: injective, surjective, bijective, none (meaning not
injective, not surjective, not bijective). For each bijective function
find its inverse.
a) f:N -> N	f(x) = x modulo 10
b) f:N -> N	f(x) = if x is odd then (x-1) else (x+1)
c) f:S -> power(S)	S is any set and f(x) = {x}
d) f:lists(S) -> power(S)	S is any finite set
	f(<x1, x2, ..., xn>) = {x1, x2, ..., xn}
	lists(S) is the set of all lists over the set S.
e) seq:N -> lists(N) is called the sequence function
	seq(n) = <0, 1,..., n>
f) dist:Axlists(B) -> lists(AxB) is called the distribute function
	dist(x, <y1, y2,..., yn>) = <<x, y1>, <x, y2>,..., <x, yn>>

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together with your SSN.