Discrete Mathematics I

Math. 125, Sec. 003, Fall 2018

Homework and Announcements

Urgent stuff

"Snowday!" Next Monday, November 19th, class will be cancelled. We will cover the problems assigned in week 12 on Monday November 26th. -- Have a wonderful Thanksgiving break.!

Please NOTE! 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 27, 2018, 9 -- 11:30 am. (ii) Putnam Math Competition Saturday, December 1, 2018, 10 am -- 1 pm (Part A) and 3 pm -- 6 pm (Part B).

Those who are interested should contact Prof. Dan Anderson (danders1"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 4:30 -- 6 pm at Math Dept. Room 4307 (open collaboration room) EXPLORATORY HALL.

General Info

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.

Homework Info

Here below are the lectures and homework assignments (HW) so far in the course.

  1. Lectures and Homework (HW) from Week 1. HW for Monday, September 10. NB! Monday Sept. 3rd is Labor Day. 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.
  2. Lectures and Homework (HW) from Week 2. HW for Monday, September 10. The Cartesian 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").
  3. Lectures and Homework (HW) from Week 3. HW for Monday, September 17. : 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! Two properties 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.
  4. Lectures and Homework (HW) from Week 4. HW for Monday, September 24. : 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.
  5. Lectures and Homework (HW) from Week 4. HW for Monday, October 1st. : There are roughly three ways to prove assertions in mathematics (1) Direct proof: To prove that an assertion/propositon P implies Q we assume P and work our way toward Q. (2) Indirect proof: Here to prove that assertion/propositon P implies Q we assume that Q does not hold and show that P does not hold. (3) Proof by contradiction: This is one of the corner stones of mathematical reasoning. Here in order to prove a proposition, we assume that it is wrong and use that (together with legitimate mathematical facts) to derive some nonsense (i.e. contradiction). Once we have a contradiction, then we conclude that our assumption that our original proposition is wrong is itself wrong. Hence the proposition is true. We saw two Greek classical examples of this way or proving, namely that the square root of 2 is irrational, and that there are infinitely many prime numbers. -- To prove assertions about the natural numbers, many times we can use induction. There are two kinds of induction, the weak form and the strong form of induction. The weak form is the form where we (i) prove that P(a) is true, and then (ii) that P(n) implies P(n+1) where n is an arbitrary natural numbers. Note, that in (ii) we are not assuming that P(n) itself is true, we only show that it implies P(n+1). Once we have shown (i) and (ii), then we can conclude that P(n) is true for all natural numbers n from and including a. -- Note that inductive proofs can many times yield procedures or algorithms on "how to do things" constructively. For example, when we proved, by induction (weak form) that any white 2n x 2n chess board, where one black square (not to be tiled) could be tiled using L-shaped tiles (consisting of 3 tiles), then we can obtain an algorithm to tile any such board. Here below is a picture of such a tiling (curtsey of Mr. Samuel Levy, a student of this class in Spring of 2014, who turned this inductive proof into an algorithm): The strong form works in a similar fashion, except that in (ii) we show that if all the propositions P(a), P(a+1),...,P(n) are true, then so is P(n+1). Recursion is what we mean when constucting something inductively. It is really the same thing as induction, but it refers more to construction, rather than proofs. For example, we can define a sequence of numbers (that is, a function from the natural numbers to the reals) in a recursive manner, then we can guess a formula for this function, and then prove by induction that the formula is actually correct for all natural numbers. We saw examples of such functions defined recursively. Sometimes it is very easy to guess what the formula is (and so we can prove it by induction). Other times we can derive a recursive definition for the function (and hence easily write a computer program to compute the value of the function for an arbitrary natural number) although it might be impossible to find an explicit formula for the function. -- Next week we will continue to discuss recursively defined functions/sequences and how possibly find formulas for them.
  6. Lectures and Homework (HW) from Week 6. HW for Tuesday, October 9. : Recursively defined number sequences are very common in computations. When a number sequence is defined recursively, it might be difficult to find a formula (as a function of n alone!) of it. As an example, we saw how the well-known Fibonacci number sequence had an elegant and natural recursive presentation, but a formula for the n-th Fionacci number involved both fractions and square roots. So from this point of view it wasn't even clear that it was an integer at all!. However, by examining general variations of Fibonacci-like number sequences, we saw that by using the characteristic equation we can use the roots to present a formula for the n-th term of the sequence. This is very easy and direct if we do not have any additional function f(n) added in the recursion (i.e. f(n) = 0 for all n). But if f(n) is not identically zero, then we must first guess a particular solution for the recursion, and then use that to find all solutions to the recursion. We will continue to discuss such "educated guesses" next week.
  7. Lectures and Homework (HW) from Week 7. HW for Monday, October 15. : We have seen some "educated" ways to guess a particular solution to a linear recurrence equation of 2nd degree. There are two things to note when guessing the form of such a particular solution. (i) If the particular solution is exponential function times a power of n, then our particular solution should be a polynomial P(n) times the exponential function. Now, the degree of the polynomial should be (a) the same as the power of n if the base of the exponential part is not a root of the characteristic polynomial, (a) one more than the power of n if the base of the exponential part is a single root of the characteristic polynomial, (c) two more than the power of n if the base of the exponential part is a double root of the characteristic polynomial. (ii) The second (and very important) part is that if we know the particular solutions for various functions, then any linear combination (sums and scalar multiplications) of these particular solutions will be a particular solution to the corrsponding recurrence equation where the function is the same combination of the other given added functions. -- Next week we will start discussing Chapter 6 on how to count possibilities.
  8. Lectures and Homework (HW) from Week 8. HW for Monday, October 29. : When counting elements in a given set, one has to proceed with some care and not be to hasty to rush to an answer. The two most fundamental ways of counting elements are given by the product rule (that the number of elements in a Cartesian product of finite sets is the product of their cardinalities) and by the inclusion/exclusion principle (I/E). The I/E principle can be generalized to count the number of elements in the union of 3, 4 and in general k sets, however, the formula we obtain for "many" sets is very involved and complicated. But, for 2 or 3 sets using the I/E principle is perfectly within reach. -- Another principle that can be used to show an existence of a sertain situation is the Pigeon Hole Principle (PHP). This principle is used when we need to prove that we have two or more elements that have a common property. We don't use this in direct counting per se. For that (exact counting) we need to first establish the number of element in some basic sets. For example n! called n-factorial is used in many formulae where we express the number of combinations of various kinds. Here , n! is the number of injective functions from the set of the first n natural numbers to itself. This is the same as the number of rearrangements of n distinct symbols/items from left to right.
  9. Lectures and Homework (HW) from Week 9 HW for Monday, October 29. : The number of ways to form an ordered r-tuple from a set of n elements is P(n,r) = n!/(n-r)!. From this we see that the number of ways to choose an r-element subset from a set of n elementsis given by C(n,r) = n!/[r!(n-r)!]. Here C(n,r) is the binomial coefficient and it is read "n choose r". The binomial coefficients are very important and come up in many combinatorial situations. For example, the number of ways to put n identical marbles into r labeled jars is given by C(n+r-1,r-1). One of the most important applications of the binomial coefficients is in the Binomial Theorem (BT), that gives a formula for expanding (x+y) to the n-th power. Pascal's Triangle can be used efficiently when n is not too large, but for large n it is much better to use the BT directly and draw our conclusions from that. -- Note, that many times it is convenient to use calculus and find derivatives when we want to derive an identity involving binomial coefficient, as we saw in class.
  10. Lectures and Homework (HW) from Week 10. HW for Monday, November 5th. : Graphs are one of the most used and important concepts in discrete mathematics. Countless discrete problems, both practical and theoretical, can be phrased in terms of graphs. There are many types of graphs; simple graphs (where there is at most one edge between two distinct vertices) and general graphs (where there could be multiple edges between two vertices, and also possible loops, that is edges with just one endvertex.) The complete graph on n vertices is a simple graph in which every pair of distinct vertices are connected by an edge. Since there are exactly n-choose-2 such pairs of distinct vertices, we see that the complete graph on n vertices has n-choose-2 edges. Every simple graph is a subgraph of the complete graph on n vertices, where n is the number of vertices of the given graph. Two important ways to form a subgraph is by (i) removing/deleting edges, and (ii) removing/deleting vertices. Note that when we remove/delete a vertex, then we also remove all the edges with that vertex as an endvertex. -- The Hand Shaking Theorem states that the sum of all the degrees of the vertices of a graph equals twice the number of edges of the graph. -- Another important class of graphs consists of bipartite graphs. These are graphs where we can partition the vertices into two groups, such that every edge connects vertices from the two groups (and no edge has both endvertices in the same group.) The complete bipartite graph is a bipartite graph where all the possible edges between the two groups are present. -- When we say that "two graphs are the same", what we really mean is that their internal structure is the same, or that they can both be represented by the same drawing. However, given two graphs, it is in general a very difficult task to determine if they are "the same" or not. The formal way of saying that two graphs are the same is to say that they are isomorphic. An isomorphism between two graphs is a bijective function/map from the vertex set of one to the vertex set of the other, such that two vertices in the former are adjacent if, and only if, their images (under the bijective function) are adjacent in the second graph. To find such an isomorphism can be a daunting task, even if we know it exists! However, there are some easy ways to determine if two graphs are not isomophic. For example, two isomorphic graphs must have (i) the same number of vertices, (ii) the same number of edges, (iii) the same number of vertices of degree k for every k, (iv) the same number of triangles, squares etc... If one of these requirements fails, then we can conclude that the graphs are not isomorphic.
  11. Lectures and Homework (HW) from Week 11. HW for Monday, November 12th. : We now come back to the question we considered when introducing graphs, namely the Bridges of Königsberg. This first problem of graph theory, that was solved by Leonard Euler around 1730, can be solved by considering the corresponding (general) graph and see if it is Eulerian or not. A graph is Eulerian (i.e. contains a circuit that covers all the edges) if and only if it is connected and the degree of each vertex is even. This theorem is what solved the problem of the Bridges of Königsberg and answered it in the negative. One direction is fairly easy; namely, if a graph has an Eulerian circuit then it must be connected and the degree of each vertex must be even. The other direction is not so easy, but we saw in class an algorithm (one of many that exist!) that will create an Eulerian circuit in a connected graph where every vertex has an even degree. Unlike the problem of determining whether two graphs are isomorphic or not, this problem of finding an Euler circuit of an Eulerian graph is computatble in a reasonable amount of time.
  12. Lectures and Homework (HW) from Week 12. HW for Monday, November 26th. : We have seen in class an algorithm to find an Eulerian circuit in a graph that is Eulerian (connected and has all degrees even). Note, determining whether a graph is Eulerian or not is easy; just check connectivity and check that all the degrees are even. Knowing that a graph is Eulerian, it is another task to find the Eulerian circuit. That can take some effort, although it is doable in a reasonable amount of time. -- Recall that a tree is a connected graph with no cycles, and a forest is a graph in which every component is a tree. There are many nice characterizations of trees; (i) A graph is a tree if and only if there is a unique path between any two vertices in the graph. (ii) A graph on n vertices is a tree if and only if it is connected and has n-1 edges. To prove this we used the fact that every tree has at least two (in fact one suffices!) leaves, where a leaf is a vertex of degree one. A spanning tree of a graph is a subgraph that (i) is a tree and (ii) contains all the vertices of the graph. Every graph contains at least one spanning tree. Of great interest is to find the minimum weight spanning tree (MWST) of a weighted graph (where we have attached a real weight to each edge). There is an elegant algorithm to find a MWST for graph for any weighted graph, called Kruskal's Algorithm. The interesting part about Kruskal's Algorithm is that we don't have to worry about keeping the subgraph at each step connected; it will connect at the very last step and yield a MWST for the graph. -- That is it for lectures folks!
  13. Lectures and Homework (HW) from Week 13.

Exam Info (Quizzes, Midterm and Final)

Here below are the handwritten solutions to the quizzes that I have handed back so far.

Below is a review sheet for the first midterm. The midterm is to be held in the classroom, Monday, October 22nd. -- Please note that this review sheet is not a recipe for the midterm, but merely a collection of sample problems you should attempt to practice your skills!

The midterm will consist of 5 to 8 problems. There are no multiple choice questions. No calculators or cheat-sheets are allowed either. You will have 50 minutes to do the midterm and I will give partial credit. Note, that you might get full credit for a problem even if your final answer is wrong, provided that I see your method is correct! Likewise, you might not get any points for a problem even if your final answer is the correct one, if I see that your method is utterly wrong! -- The material covered in this midterm will basically be what we will have covered up until and including last week Wednesday October 10th. but not what is covered this week Chapter 2, sections 2.1, 2.2, 2.3, 2.4, 2.5. Chapter 3, sections 3.1, 3.2. Chapter 4, sections 4.1, 4.2, 4.3. Chapter 5, sections 5.1, 5.2.

The review sheet is only intended to give you an idea of what might be on the midterm and I don't plan on going over them in detail before the midterm. But if you have some quick questions, I could give you a hint.

Dear folks! Make sure you get back your midterm exam. The median score was 59/100. -- Please note that the curve I present in class is only how you did in this midterm amongst yourselves. It has nothing to do with your final letter grade for the course since I only record the *number* grade. I will not record the curved letter grade for this midterm exam.

Here below are handwritten solutions to the midterm exam. Please print out a hard copy for your records. Make sure you understand the solutions. -- If you have any concerns, you are welcome to come and discuss them with me, but read the solutions first.

EOF/EOS