Discrete Mathematics I
Math. 125, Sec. 003, Fall 2017
Homework and Announcements
All handouts for the class will be posted as pdf files
after a bullet sign. All these handouts are copyright
material and are intended for the students of this class only.
IN PARTICULAR, DO NOT POST THESE HANDOUTS ON ANY OTHER
WESITE OR INTO AN ARCHIVE DATABASE!
GMU Undergraduate Math competitions Activities
Two upcoming competitions are:
(i) Virginia Tech Math Competition Saturday,
October 21, 2017, 9 -- 11:30 am.
(ii) Putnam Math Competition Saturday,
December 2, 2017, 10 am -- 1 pm (Part A) and 3 pm -- 6 pm (Part B).
Those who are interested should contact either
Prof. Dan Anderson (danders1"the-at-sign"gmu-dot-edu), or
Prof. Jim Lawrence (lawrence"the-at-sign"gmu-dot-edu) for
more info. -- Recruitment is underway, and practice for
the above competitions will be:
Friday Putnam Practice (aka "Chill Math")
most Friday's at 4:45 -- 6 pm at Math Dept. Room 4307 (open collaboration room)
Here below is a pdf version of the Syllabus.
It outlines the overall plan of
the course and should answer all practical questions, like when tests are,
what the grading policies are etc. If you didn't get a hard copy from me,
then please print one for your records.
Here below are the lectures and homework assignments (HW) so far in the course.
- Lectures and Homework (HW) from Week 1.
HW for Monday, September 11.
A set is a well defined collection of objects
that we call elements. When dealing with sets
we usually agree on a given universe in which
we "live" and work. For the most part we will be dealing
with sets containing numbers of some sort (i.e. integers,
rational numbers, real numbers etc.) When we have two sets,
then we can form their intersection, union,
difference, symmetric difference and complement
(we do need the universe to form the complement).
To verify or prove identities involving sets, one can use
containment tables to list all the possibilities of an arbitrary
element being in a given set (denoted by 1) or not being
in the set (denoted by 0). This method is quite good
when the number of sets is three or less. However, when
the number of sets is large, then it is better to use
basic identities to prove that one set is equal to (or contained
in) another. Note that Venn diagrams are good to get
a sense of how sets relate to one another, but it is not
a proper way of proving things! -- For each set we can
attach either a non-negative integer or "infinity" for
the number of elements in the set. If a set has n
elements, for some non-negative integer n, then
we say that the set is finite, otherwise we say
the set is an infinte set. Although we will not
delve into infinite sets, note that there are different
notions of infinity. For example the set of the real numbers
is "more infinite" than the set of natural numbers. -- For
a given set A we can obtain the power set
of A, which contains all the subsets of A.
If A is finite and contains n elements, then
the power set is also finite and contains 2n
elements. The Cartesian product of sets is the collection
of all ordered tuples, where the i-th coordinate is an
element of the set number i. If all the sets are the same
(for example the real number line), then
we obtain familiar sets of points, like 2-tuples representing points in
the Eculidean plane, and 3-tuples representing points
in the three dimensional real space.
- Sec 2.1: 1, 3, 9, 10.
- Sec 2.2: 1, 4, 5, 8, 10, 13, 18, 20, 22, 26, 30.
- Some additional problems from week 1.
- Lectures and Homework (HW) from Week 2.
product of finite sets is also finite
and its number of elements is (nicely so)
the product of the number of elements of the original finite sets
we started with. -- A binary relation from a set A
to a set B is simply a subeset R of
the Cartesian product A x B. This is a pretty general
definition, but within this framwork there are many interesting
scenarios representing relationships between elements of either
different type or the same type (when B is also
the set A). Hence, any binary relation is determined
by some such subset R. Some conditions on the binary
relations are of interest; (i) reflexivity, (ii) symmetry,
(iii) antisymmetry and (iv) transitivity are four very
important properties for binary relations. Depending on how
the binary relation is defined, we must check to see if the
binary relation at hand saitifies any of these conditions. If
a binary relation satifies (i), (ii) and (iv) then we say
it is an equivalence relation, and we say that two
related elements are "equivalent" to one another. (Since clearly
the "equal to =" relation is clearly an equivalence relation, one
can view an equivalence relation as a generalization of being "equal to").
Equivalence relations have a very nice description: Every equivalence relation
on a set A yields a partition of A (where each part
contains exactly all elements related to one another) and vice versa,
every partition on A yields an equivalence relation (given
by "two elements are equivalent if they are contained in the same part").
- Lectures and Homework (HW) from Week 3.
HW for Monday, September 18. :
So if we have an element a, then the unique part in which a
is contained in, is called the equivalence class of a.
The collection of all equivalence classes (or the distinct parts in the
corresponding partition) is called the quotient set.
As we saw in class, it is possible to have an infinite set,
where each equivalence class is infinite and where there
are only finitely many equivalence classes (so the quotient set
is finite!). -- Another type of binary relation that is of great importance
is a partial order, which works the same way
as the relation "less than or equal to" on numbers. A binary
relation is a partial order if it satisfies (i), (iii) and (iv)
from above, that is: reflexivity, antisymmetry and
transitivity. A set that is provided with a partial order
is calle a partially ordered set or just poset
for short. Clearly the relation "less than or equal to" is a partial
order, but the catch with general partial orders is that given
two elements in a poset, they might not be comparable; one is
not necessarily greater than the other. As an example, we considerd
the "subset of or equal to" among subsets of a given set. This
turns out to be a partial order, but given two sets, one is not
necessarily contained in the other (in fact, they could be disjoint!)
If every two elements in a poset are comparable (so one is always
greater than or equal to the other) then the poset is said
to be totally or linearly ordered. This means
essentially that we can list all the elements in a row according to
size. In general, when working with posets it is a good idea
to look at their Hasse diagrams. Again, just like with Venn diagrams,
the Hasse diagrams don't prove anything, but they give you an idea
of the structure of the poset at hand. Not all posets have maximum
or minimum elements, but every finite poset has at least one maximal
element and at least one minimal element. -- Chapter 3 deals with
functions, something most of you have some familiarity with. Perhaps
what you are not used to is to specify the domain and
the codomain (or target) of a given function.
By the image of a function we mean all the elements
in the codomain that are a image of the function. The image
of a function does not have to be the whole codomain!
of a function are of great importannce: (i) a function
is one-to-one (1-1) or injective if distinct
elements in the domain are mapped to distinct elements of the codomain.
(ii) A function is onto or surjective if the image
of the function is the whole codomain, that is, every element in
the codomain/target actually is an image of some element of the domain.
If a function is both injective and surjective, then it is bijective.
We can compose functions, and the composition does satisfy some nice
properties. For example, composition of functions is associative
in the sense that it does not matter the order you evaluate the composition,
as long as the left-to-right order is intact. Note that composition
is not commutative: f composed with g is not
the same as g composed with f! So we must take
some care! Bijective functions are very nice, since these functions are precisely
the ones that have inverses, that is functions that "cancel" the effect
of the function (i.e. you can get everything back in the domain from the codomain.)
We will see many examples of this in class and problems.
- Sec 2.3: 3, 5, 7, 9, 11.
- Sec 2.4: 2, 3, 5, 6, 7, 10, 11, 12, 17, 20, 21.
- Some additional problems from week 2.
- Lectures and Homework (HW) from Week 4.
HW for Monday, September 25. :
Arithmetic with integers are of vital importance in computer science.
What makes the integers hard to deal with is, in part, due to
the fact that one cannot always divide one integer into another
and obtain an integer as an output. Division must be done with more
care. When dividing one integer into another we obtain two
numbers; the quotient and the remainder
using the well-known division algorithm. --
The Fundamental Theorem of Arithmetic states that
every natural number has a unique prime factorization. Using the
prime factorization we can immediately spot if one integer divides
another; it happens when all the prime factors of the denumerator
occur with the same or less power as in the numerator. Also the
greatest common divisor (gcd) and the least common multiple (lcm)
of two integers can be easily computed from the prime factorizations.
However, it is a very hard computational problem to find/obtain the
prime factorization of an integer. Luckily there are better ways
to compute both the gcd and lcm of two integers. The
Euclidean Algorithm (EA) is one of the oldest and yet fastest
ways to compute the gcd of two integers. In fact it is so fast that
no faster way has been developed in thousands of years!
Using the EA we can solve an integer equeation of the form
ax + by = c for arbitrary fixed integers a,b,c.
Using the EA we first determine if there is a solution at all.
If that is the case, the we use the EA to find one solution.
Once we have one solution, then there are infinitely many solutions
and we have a formula for writing down all the solutions to
this integer equation. -- Next week we will start on induction,
and various methods of proving.
- Sec 2.5: 3, 4.
- Chapter-2 Review: 7.
- Lectures and Homework (HW) from Week 4.
Exam Info (Quizzes, Midterm and Final)
Here below are the handwritten solutions to
the quizzes that I have handed back so far.