JDEP 183H
Homework
Assignment 1A
Sets,
Relations, and Functions
Assigned: September 5, 2006
Due: September 19, 2006
(12:30 PM)
(Homework 5
minutes late will not be accepted)
Set Basics
1. (10 points) List all members of the following sets.
(a) 
(b) 
(c) 
where 
is the set of all
positive numbers
(d) 
where 
is the set of all
positive numbers
(e) 
where is the set of all
positive numbers
2. (10 points)
Consider the following six subsets of Z. (Definition:
Z = the set of all integers = ( …, -3, -2, -1, 0, 1, 2, 3, …}.)
A = { 2m + 1 | m Î Z }
B = { 2n + 3 | n Î Z }
C = { 2p -3 | p Î Z }
D = { 3r + 1 | r Î Z }
E = { 3s + 2 | s Î Z }
F
= { 3t - 2 | t Î Z }
Which
of the following statements are true and which are false? Show your reasons.
(a) A = B
(b) A = C
(c) B = C
(d) D = E
(e) D = F
(f) E = F
Set Operations
3. Application of
Knowledge (14 points)
Consider an alphabet Σ = { a, b }. It has only two letters. A word (or w) is any finite
string of letters that you can form out of the alphabet. For example, abba is a word formed
using only the letters from the alphabet Σ. The collection of all words is called a language,
and is denoted as 
. For example, = { ε, a, aa, ab,
ba, bb, aaa, aab, abb, bbb, bba, baa, …}.
And is an infinite
set! The symbol ε denotes the empty
word or null word.
Now, suppose Σ = { a, b }; A
= {a, b, aa, bb, aaa, bbb }; B = { 
: length(w) ≥ 2}; and C = { : length(w) ≤ 2}. (Hint: Don’t forget the empty word ε,
which has a length of zero!) List all
members of the following:
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) power set of Σ
Binary Relations
4. (18 points)
Determine whether or not each of the binary relations 
defined on the given
sets A are reflexive, symmetric, or transitive. If a relation has a certain property, prove
this is so; otherwise, provide a counterexample to show that it does not. Determine also whether a relation is equivalence.
(a)
A = { 3, 4 };
= { (3, 3), (4, 4)}
(b)
A is the set of all humans; 
if and only if 
where means “a is the
parent of b.”
(c)
A is the set of cross products of integers, Z×Z;
if and only if 
is odd.
(d)
A is the set of cross products of integers, Z×Z;
if and only if 
.
(e)
A is the set of all humans; if and only if 
where means “a and b
are both fans of the same musician.”
Assume that a human may be a fan of more than one musician.
(f)
A is the set of natural numbers, N; 
if and only if 
is an integer.
5. (8 points) Given the following relations:
.
.
.
.
List 4 members for each of the following sets; if not possible, explain.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
Functions
6. (15 points) Determine whether each of these functions is
onto and one-to-one (where Z is the set of integers, and R is the set of real numbers).
Explain or show an example to support your answers. (Hint:
For (e)-(g), think carefully about the domain, target, and range
involved.)
(a)
f: Z → Z , 
(b)
f: Z → Z , 
(c)
f: R → R , 
(d)
f: R → R , 
(e)
f: Z → Z , 
(f)
f: R → R , 
(g)
f: R → Z ,
7. (15 points) We define functions mapping R to R
(where R is the set of real numbers) as follows: 
, 
, 
. Find
(a) 
(b) 
(c) 
(d) 
(e) 
8. (15 points) Find the inverses of the following functions mapping R to R (where R is the set of real numbers):
(a) 
(b) 
(c) 
(d) 
(e) 
where 
is a constant.
* Based on (Goodaire and Parmenter 2002), (Grimaldi 2003), and (Ross and Wright 1988).