Then we may define: Line segment: that portion of a Une contained between two given points on a Une. Tlie words in this definition other than point, line, and between are without special meanings and thus may be used freely. Our use of undefined words is the first phase of our abstraction of mathematics from physical reality.

The penciled line on our paper and the chalk line on our blackboard are physical realities, but line, the undefined mathematical concept, is something quite apart from them. In geometry we make statements about a line which we shall call axioms which correspond to observed properties of our physical lines, but if you insist on asking: "What is a line? These statements will be ordinary declarative sentences which are so precisely stated that they are either true or false. We will exclude sentences which are ambiguous or which can be called true or false only after qualifications are imposed on them.

The following are acceptable statements:. All triangles are isosceles If. At the very beginning we must choose a few statements which we will call "true" by assumption; such statements are calledj' axi- oms. They are not statements about the properties of the physical world. Since mathematical theories can begin with any set of axioms at all, they are infinite in their variety; some of them are interesting and useful, others merely interesting, and still others only curiosities of little apparent value.

The choice of a set of axioms which leads to an interesting and useful theory requires great skill and judgment, but for the most part such sets of axioms are obtained as models of the real world. We look about us, and from what we see we construct an abstract model in which our undefined words correspond to the most important objects that we have identi- fied, and in which our axioms correspond to the basic properties of. The mathematics which you will use as a scientist is entirely based on axioms which were derived in this fashion.

From our set of axioms which we have assumed to be true we now proceed to establish the truth or falsehood of other statements which arise.

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We shall not dwell upon these Laws of Logic here, but if you are interested you can read about them in the References given at the end of the chapter. Except for a few tricky places Avhich we will discuss below, you can rely upon your own good sense and previous experience in logical thinking. Whenever doubts arise, however, you must refer back to the full treatment of these logical principles. When we have shown that the truth of a given statement follows logically from the assumed truth of our axioms, we call this statement a "theorem" and say that "we have proved it.

The main business of a mathematician is the invention of new theorems and the construction of proofs for them. The discovery of a new theorem depends upon deep intuition and intelligent guessing, and the process of making such a discovery is very much like that of creative effort in any field.

After our intuition has led us to beheve that a certain statement is true, we must still prove it; and this is where our use of logical deduction comes in. Since we shall need to have a good understanding of the nature of proof, we will devote the rest of this chapter to a discussion of various prob- lems which you will meet in mathematical proofs. Negations Whenever we make a statement about mathematics or anything we mean to assert that our statement is true.

As you will see shortly, there are times in mathematics when we wish to assert that a given statement is false. Thus we may say:. Rather than using the awkward "It is false that Let us put this a little more formally. We use p to represent a given statement and not-p to represent its negation. Definition: The negation not-p of a given statement p is a statement such that: a If p then not-p is false.

In many cases you can form negations easily by inserting a "not" in a convenient place, but in other cases you must be more subtle. There are general rules for taking negations which you can find in the References listed at the end of this chapter, but for our present pur- poses we shall rely on some examples and your own good sense. Illustrations 1. Illustrations 3 and 4 are rather deceptive, and you should think them through carefully to be sure that you understand why these are the correct negations.

Negations of this type are particularly important in mathematics. Choose an ordinary, nontechnical word, and build a circular chain of defini- tions from this word back to itself.

## College of Arts and Sciences

Use a standard dictionary for your defini- tions. Do not put simple connections such as "the," "and," "in," "is," etc. In any standard dictionary look up the definitions of a mathematical "point" and "line. In Probs. Then define: 3. Isosceles triangle. Write the correct definitions.

Intersect for lines : Two lines are said to ititersect if and only if they have one or more points in common. Parallel for line segments Two : line segments are said to be parallel if and only if they do not intersect. Congruent for triangles : Two triangles are congruent if and only if the angles.

Concentric for circles : Two circles are concentric if and only if their radii are equal. Complete each of these statements to an unambiguously clear true statement by prefixing one of the following phrases: "for all X," "for some x," "for no x. The base angles of a given triangle are equal. Angle yl is a right angle. The sum of the interior angles of a given square is For every pair of similar triangles, Xi and X2, Xi is congruent to xj.

For all triangles x, the sum of the interior angles of x is equal to Implications At nearly every turn in your study of mathematics, you will meet statements of the form: "If then. Let us start from a given implication: "If p, then q"; and now sup- pose that we interchange the two statements, p and q. We obtain a new implication: "If q, then p," which is closely related to the given implication but which is surely different from it. Definition: The implication: "If q, then p" is the converse of the imphcation: "If p, then q.

Indeed, the con- verse of a true imphcation may be true or false; examples of each kind are given below. True Converse: If two triangles are similar, then they are congruent. False 2. Remember: The truth of an implication docs not imply the tnith of its converse. There are occasions, of course, where "If p, then q" and "If q, then p" are both true.

In these circumstances we say that p and q are eciuivalent. The converse an implication is often confused with its contra- of positive, which another implication defined in the following fashion: is. As before, we start from a given implication, "If p, then q"; and now we do two things: 1 Ave take the negation of each of the statements p and q and thus obtain new statements 7iot-p and not-q; 2 then we interchange the two statements ''not-p and not-q.

Definition: The implication: ''If not-q, then nof-p'' is the contraposi- tive of the implication: "If p, then q. The big difference between the converse and the contrapositive of an implication is a result of the following law of logic:. Law of Logic. An imphcation and its contrapositive arc either both true or both false. As we shall see below, contrapositives can be very helpful to us. When we find it difficult to Drove that an implication is true,we can form its contrapositive. If Ave can prove this to be true, we have automatically established the truth of the given implication.

Consider the implication "If a polygon is a square, then it is a rectangle. On the other hand, a polygon cannot be a square unless it is a rec- tangle. Or "In order that a polygon be a square, it is necessary that it be a rectangle. The folloiving three statements all carry the same meaning: If -p, then q p is a sufficient condition for q.

By examining the Recapitulation above you can verify that the first of these is equivalent to "If p, then g," whereas the second is equiva- lent to the converse "If q, then p. When the propositions p and q are equivalent, l oth "If p, then q" and "If q, then p" are true. In this case we say that. There is another way of expressing these same ideas. By interchanging p and q in statements 1 and 2 above we see that the converse of the above implication can l e written in two ways:. Note, then, that the substitution of "only if" for "if" in an implica- tion changes the implication into its converse.

We summarize this discussion with a table, in which entries on the same horizontal line are equiv;dent statements. The first set of lines. If a is divisible by 3, then 2a is divisible by 6. If the sides of a triangle are all equal, then the triangle is equiangular. If a quadrilateral is a parallelogram, then its diagonals bisect each other. Write the contrapositive of the converse of "If p, then q. Write the converse of the contrapositive of "If p, then q. Write a true implication whose converse is true.

Write a true implication wliose converse is false. If the base angles of a triangle are equal, the triangle is isosceles. If two triangles are congruent, their corresponding altitudes are equal. If two lines are perpendicular to the same line, they are parallel. If two spherical triangles have their corresponding angles equal, they are congruent.

If a triangle is inscribed in a semicircle, then it is a right triangle. If a body is in static equilibrium, the vector sum of all forces acting on it is zero. If a body is in static equilibrium, the vector sum of the moments of all forces acting on it is zero. If two forces are in equilibrium, they are ecjual, opposite, and collinear.

If three nonparallel forces arc in ctiuilibrium, their lines of action are con- current. The implication of Prob. Give two answers to each problem. Two if and only if they are equidistant. An integer and only if it is divisible by 2. Three concurrent forces are in equilil rium if and only if their vector sum is. A lever is balanced if and only if the algebraic sum of all moments about its. She sued for breach of promise.

Can she logically win her suit? Direct Proof Most of the proofs you encounter follow a familiar pattern will which is The simplest type of such a proof called "direct proof. Special fact of the given problem 2 If p is true, then q is true. An axiom, definition, or previously proved theorem Conclusion: q is true.

## Courses | Department of Mathematics

In more complicated situations this secjuence may be repeated. Prove that the square of an odd number i. Given: 1 x is an odd number. Definition of odd number Conclusion: x- is odd. In constructing such proofs you will have to choose the appropriate "previously proved theorems" and arrange them in a suitable order. There is no automatic way of doing this; you must develop skill through experience and the use of your originality. Other Methods of Proof a Indirect Proof. If you have difficulty in constructing a direct.

The method of "indirect proof" rehes on the fact that, if not-p is false, then p is true. Hence, to prove that p is true, we attempt to show that not-p is false. The best way to accomplish this is to show that not-p. In other words, we add not-p to the list of given statements and attempt to show that this. When the contradiction is reached, we know that 7iot-p is not consistent with our given true statements and hence that it is false. Hence p is true. To illustrate indirect proof let us consider a familiar theorem in geometry. Prove : if two lines are cut by a transversal so that a pair of alter-.

Conclusion: AB CD. Consequently AB CD. Use of the Contrapositive. When we are trying to prove the 6 truth of an implication "If p, then q," we can just as well prove the contrapositive " If not-q, then not-p. Often there are great similarities between indirect proof and the proof of the contrapositive.

Let us consider the theorem of Illustration 1. The contrapositive of the implication stated in Illustration 1 is: obtained by cuttmg "If two lines are not parallel, then the alternate interior angles these lines by a transversal are not equal. Hence the given implication is true. Proof of Existence. Before you spend a lot of time and c money a good on a high-speed computer, say trying to solve a problem, it is. You have probably never seen problems that do not have solutions, formost textbooks and teachers consider it to be bad form to ask students to do something which is impossible.

In actual prac- tice, however, such problems may arise and it is a good idea to know. A very simple example of such a problem is the following: Find all the integers. In order to reassure you that you are working on problems that do have solutions, mathematicians have developed a number of "existence theorems. There exists a number x which has a given property. If a and b are any real numbers such that a? Although there are other forms of existence proofs, a constructive proof of this kind is considered to be of greater merit, and this method is used widely in establishing the existence of solutions of various types.

Methods of Disproof If you have tried unsuccessfully to prove a conjectured theorem, you may well spend some time trying to disprove it. There are two standard methods for disproving such statements. In this case we assume that the given statement is true and then proceed to derive consequences from it. If we succeed in arriving at a consequence which contradicts a known theorem, we have shown that the given statement is false. Disprove the statement: "The square of every odd number is. But let us disprove it from first principles. Both sides are supposed to represent the same integer, but the left hand side is not divisible by 2, while the right hand side is divisible by 2.

This is surely a contra- diction, and so the given statement is false. This method is effective in dis- proving statements of the form:. For all values of x, a certain statement involving x is true.

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In order to disprove such an assertion, we proceed to find a "counter- example. Disprove the statement: "The square of every odd number is even.

Weclose with this warning: Although disproof by counterexample is a valid procedure, theorems are not to be proved by verifying them in a number of special cases. Be sure that you do not confuse these two ideas. Prove those which are true, and dis- prove those which are false. The sum of two even integers is odd. The product of two even integers is a perfect square.

Two triangles are congruent if two sides and the angle opposite one of these of one triangle are equal, respectively, to the corresponding parts of the other triangle.

If two triangles are similar, then they have the same area. If the vector sum of all the forces acting on a body is zero, then the body is in. If the vectorsum of the moments of all the forces acting on a body is zero, then the body is in static equilibrium. The sum of the exterior angles of any triangle is equal to Any two medians of a triangle bisect each other. You are given the following axiom: "One and only one line can be drawn through any two points. You are given the theorem: "At most one circle can be drawn through three, distinct points. Give an indirect proof of the theorem: "There exist an infinite number of primes.

This theorem is due to Euclid. Mathematical Models By this mean by time you should have begun to understand what we saying that mathematics Mathematical proof is a process is abstract. Scientists, however, spend their lives uncovering the secrets of nature, and engineers put these discoveries to work for the benefit of our society. The key to this matter is the concept of a "mathe- matical model" of nature. The first step in the study of any branch of science is that of observing nature.

When enough facts have been collected, the scientist begins to organize them into some pattern. In quantitative sciences like astronomy, chemistry, and physics this pattern is expressed in terms of mathematics. The undefined terms of the abstract mathematics points, lines, etc. Then mathematical equations involving these con- cepts are used as axioms to describe the observed behavior of nature.

All of these, taken together, constitute ourmathematical model. This model, of course, is it differs from nature only a picture of nature; just as a model of an aircraft differs from the real plane itself. But just as a great deal can be learned about a plane from a model which is studied in a wind tunnel, we can use our mathematical model to help us understand nature. From our axioms, we can deduce theorems, which are true only in our abstract sense.

Nevertheless, if our model is well constructed, these theorems will correspond to observable prop-. At the very worst, these theorems are intelligent guesses about nature and serve as guides for our experimental work. At their best, when the model is a good one as is the case in most physical sciences, our mathematical results can almost be identified with physical truth. In those portions of science which you are likely to be studying along with this book, this correspondence is so close that you may not realize the difTerence between mathematics and nature itself.

It is our hope that the study of this chapter will have helped you to appreciate this important distinction. Allendoerfer, C. Oakley: "Principles of Mathematics," pp. Copi, Irving M. Courant, R. Robbins: "What Is Mathematics? Suppes, P. Tarski, A. Introduction Since numbers are basic ideas in mathematics, we shall devote this chapter to a discussion of the most important properties of our number system. We do not give a complete account of this subject, and you are likely to study it in more detail when you take more advanced courses in mathematics.

Numerous suggestions for further reading are given at the end of the chapter. Let us retrace briefly the development of numbers as it is usually presented in schools. As a j'oung child you first learned to count, and thus became acquainted with the natural numbers 1, 2, 3,. To handle such situations, fractions were intro- duced, and the arithmetic of fractions was developed. It should be noted that the invention of fractions was a major step in the development of mathematics. In the early days many strange practices were followed.

The Babylonians con. Leonardo of Pisa also called Fibonacci , whose great work Liber Abaci was published in a. The entire collection consisting of the positive and negative integers zero and the positive and negative fractions is called the system of rational nu7nbers. The advantage of using this system in contrast to the system of purely positive numbers is that it is possible to subtract any rational number from any rational number. With only positive numbers available, 3 5, for instance, is meaningless.

It is interesting to note that it took many years before negative numbers were permanently established in mathematics. Although they were used to some extent by the early Chinese, Indians, and Arabs, it was not until the beginning of the seventeenth century that mathematicians accepted negative numbers on an even footing with positive numbers. Instead, they are written as infinite decimal expansions such as 1.

The decimal. These, however, repeat after a certain point, whereas the irrationals do not have this property. The collection of all these, the rationals plus the irrationals,is called the system of real numbers. It is quite a completely satisfactory definition of a real number, difficult to give but for our present purposes the following will suffice:. Definition: A real number is a number which can be represented by an infinite decimal expansion. If you wish a more subtle definition of a real number, read Courant and Robbins, "What Is Mathematics? We give this property the name "closure" and write the following law.

Closure Law of Addition. You are very familiar with the fact that the order of addition is not important. To describe this proi erty, we say that addition is "commutative" and write the following law. To describe this property, we say that addition is "associative" and write the following law. Therefore we make the following definition:. We now prove a theorem which illustrates the fact that the sum of three real numbers is the same regardless of the order in which the addition is performed.

Theorem 1. In a similar fashion we can define the sum of four real numbers. As before, the commutative and associative laws show that this addition does not depend upon the order in which the addition is carried out. The number zero plays a special role in addition; the sum of zero and any real number a is a itself:. Since this leaves a identically as it was before the addition, we lay down the following definition. Definition: The real number zero is called the identity element in the addition of real numbers.

Definition : The real number a is called the additive inverse of the real number a. This statement is equivalent to. We now wish to retrace our steps and return to 0; hence we must add a to a. The operation of adding a undoes the operation of adding a and thus is said to be the inverse operation. We must further define the difference of two real numbers. Other cases, however, must be treated, and we include these in the definition below. We introduce the symbol a b to denote sub- traction and define it as follows. Definition Let a and : b be two real numbers.

Then by definition. You minus sign is used in two distinct ways: will notice that the 1 denotes the additive inverse of a; 2 a h denotes the differ- a ence of a and b. We shall have frequent occasion to refer to the absolute value of a real number a. This notion of absolute value is particularly helpful when we wish to obtain rules for the addition of Elementary two signed numbers. The rules for the addition of signed numbers are given by the following theorem, whose proof is left to the Problems. Theorem 2. Let a and b be two real numbers, neither of which is. Then 1 If a and b have the same sign,.

Illustrations 2. This may be justified by the computation. Multiplication of Real Numbers Now that the essential laws of addition are before us, the laws of multiplication are easy to learn; they are almost the same, with "product" written in the place of "sum. Closure Law of Multiplication. The product a X h oi any two real numbers is a unique real number c.

We now ask: "What is the identity element for multipHcation? In other words, multiplication by b leaves a unchanged, just as in addition the addition of to a leaves a unchanged. Clearly the correct choice for the identity element is 1. Definition: The real number 1 is called the identity element in the multiplication of real numbers. Compare this closely with the notion of an additive inverse above. Hence has no multiplication inverse. These are illustrations of the following law.

Distributive Law. This law is the basis for many famihar operations. For example, the usual way of multiplying 15 X 23 is. As a more complicated example, consider the following illustration. The distributive law has a number of important consequences. The first of these states the multiplicative property of zero. Theorem 3. A second consec[uence of the distributive law is the set of rules for multiplying signed numbers. Let us look at some special cases.

Evaluate 2 X Evaluate 4 X 5. These illustrations suggest the following theorem whose proof is. Theorem Let a and h he positive real numbers. Finally we wish to define division. Then the. The letters a, h, c stand for arbitrary real numbers. Distributive Law Rll. They should be carefully memorized. In more advanced mathematics these are taken to be the axioms of an abstract system called a " field.

Addition In Probs. Model your proofs on the one given for Theorem 1.

### See a Problem?

Find the additive inverse of each of the following:. Multiplication Formulate a definition for a Y. Assuming that a X X c has been defined Prob. Find the multiplicative inverse of each of the following:. Subtraction and Division Does the commutative law hold for the subtraction of real numbers? Does the commutative law hold for the division of real numbers? Does the associative law hold for the subtraction of real numbers? Does the associative law hold for the division of real numbers? Is there an identity element for subtraction? Is there an identity element for division?

If so, what is it? Proofs In Probs. You may use Rl to Rll as given axioms. Theorem 2, Sec. Theorem 4, Sec. Let "addiplication" be defined with symbol O as follows:. Under addiplication are the real numbers closed? Is addiplication commuta- tive; associative? Is there an identity; an addiplicative inverse? Exercise A. Let US look at the other laws. The natural numbers cannot satisfy since fractions of the form. V, i, etc. Exercise B. The natural numbers, however, do have several properties which are not shared by all the real numbers. The first of these has to do with their factorization.

We recall the following definition. Definition: A natural number is called prime if and only if it has no natural numbers as factors except itself and 1. For special reasons 1 is usually not considered prime. In factoring a natural number like GO, we may write. Notice that these two sets of prime factors of 60 are the same except for their order. This illustrates a general property of the natural numbers which is stated as a theorem.

Theorem 5. Unique Factorization Theorem. A natural number can be expressed as a product of primes in a way which is unique except for the order of the factors. We omit the proof of this theorem. The natural numbers have an additional property which is essential for many portions of mathematics.

## Class Schedule: Fall 12222

To illustrate this process, suppose that we try to prove Theorem 6. This works too. Similarly we can verify the formula for any value of n. But how can we prove it in general? To do this, think of the natural numbers as represented by the rungs of an infinitely long ladder based on the ground and reaching to the sky. The bottom rung is 1, the next 2, and so on. We wish to climb this ladder to any desired rung. To do so, there are two essential steps: I We must get our foot on the bottom rung. II We must be able to climb from any rung to the next rung. Clearly, if we can do these two things, we can climb as far as we please.

To proceed upward, we need a general process which will show us how to proceed from rung to rung. We start with. Suppose by some means we have reached the kih rung, for k any natural number. This is the required result. Topics include discrete probability, mathematical logic, linear algebra, and graph theory. Same as CSE The one credit UB Seminar is focused on a big idea or challenging issue to engage students with questions of significance in a field of study and, ultimately, to connect their studies with issues of consequence in the wider world.

Essential to the UB Curriculum, the Seminar helps transition to UB through an early connection to UB faculty and the undergraduate experience at a comprehensive, research university. This course is equivalent to any offered in any subject. Students who have previously attempted the course and received a grade of F or R may not be able to repeat the course during the fall or spring semester. The three credit UB Seminar is focused on a big idea or challenging issue to engage students with questions of significance in a field of study and, ultimately, to connect their studies with issues of consequence in the wider world.

Essential to the UB Curriculum, the Seminar helps students with common learning outcomes focused on fundamental expectations for critical thinking, ethical reasoning, and oral communication, and learning at a university, all within topic focused subject matter.

The Seminars provide students with an early connection to UB faculty and the undergraduate experience at a comprehensive, research university. Geometry and vectors of n-dimensional space; Green's theorem, Gauss theorem, Stokes theorem; multidimensional differentiation and integration; application to 2- and 3-D space. Third-semester calculus course for honors students and students with an excellent record in previous calculus courses. Analytic solutions, qualitative behavior of solutions to differential equations. First-order and higher-order ordinary differential equations, including nonlinear equations.

Covers analytic, geometric, and numerical perspectives as well as an interplay between methods and model problems. Discusses necessary matrix theory and explores differential equation models of phenomena from various disciplines. Uses a mathematical software system designed to aid in the numerical and qualitative study of solutions, and in the geometric interpretation of solutions. Linear equations, matrices, determinants, vector spaces, linear mappings, inner products, eigenvalues, eigenvectors. Develops the student's ability to read, comprehend and construct rigorous proofs.

Cardinals, ordinals, order-types, and operations on them. Axiom of choice. Euclidean and non-Euclidean geometries. Studies the Hilbert postulates and various models, emphasizing Euclidean and Lobachevskian geometries. Computing now plays an essential and ever-expanding role in science and mathematics. This course provides a broad introduction to computing in the sciences and in both abstract and applied mathematics. It is accessible to students early in their undergraduate program, thereby opening the door to the profitable use of computation throughout the junior and senior years.

Permutations, combinations, and other problems of selecting and arranging objects subject to various restrictions; generating functions; recurrence relations; inclusion-exclusion theorem. Theory of graphs: Eulerian and Hamiltonian circuits; trees; planarity; colorability; directed graphs and tournaments; isomorphism; adjacency matrix; applications to problems in communication, scheduling, and traffic flow. Seminar based around a specific topic or area of mathematics appropriate to juniors in mathematics and the mathematical sciences.

Open discussion during the sessions is a key feature. A first course in probability. Introduces the basic concepts of probability theory and addresses many concrete problems. A list of basic concepts includes axioms of probability, conditional probability, independence, random variables continuous and discrete , distribution functions, expectation, variance, joint distribution functions, limit theorems.

Rigorous derivation of statistical results, clarification of limitations of statistical analysis, extensive use of computational software, application of statistical methods to case studies. Topics include: Graphical and numerical techniques for exploring data. Use and accuracy of population samples using parametric and nonparametric methods. Determination of probability distributions from statistical data. Use of computational methods based on resampling of data to determine reliability of statistical information.

Classical statistical inference methods: probability distribution estimation, confidence intervals for statistical results, hypothesis testing for statistical significance. Fitting of data using linear regression and determining the accuracy of fit. Bayesian methods for estimating probability distributions using prior information.

Advanced topics such as importance sampling for understanding the probability of rare events. Informal and formal development of propositional calculus; predicate calculus and predicate calculus with equality; completeness theorem and some consequences. For math majors in Concentration C, and majors of science and engineering. Surveys functions of several variables; differentiation, composite, and implicit functions; critical points; line integrals; Green's theorem. Vector field theory; gradient, divergence, and curl; integral theorems. Introduces functions of a complex variable; curves and regions in the complex plane; analytic functions, Cauchy-Riemann equations, Cauchy integral formula.

Surveys elementary differential equations of physics; separation of variables and superposition of solutions; orthogonal functions and Fourier series. Introduces boundary value problems, Fourier and Laplace transforms. A theoretical introduction to the basic ideas of modern abstract algebra. Topics include groups, rings, fields, quotient groups and rings, and the fundamental homomorphism theorems. Also may include applications to number theory.

For mathematics, science, and engineering majors with strong mathematics backgrounds.

Theory of Fourier series and transforms, orthogonal sets, special functions, applications. For students of physics, electrical and other areas of engineering, and mathematics. Analyticity; calculus over the complex numbers. Cauchy theorems, residues, singularities, conformal mapping. Weierstrass convergence theorem; analytic continuation. Weierstrass and Mittag-Leffler theorems, harmonic functions, conformal mapping and Green's function, analytic equivalence, and Riemann's mapping theorem.

Montel's theorem, external mappings. Metric spaces and abstract topological spaces. Continuous functions and homeomorphisms. Subspace, product, and quotient topologies. Separations axioms. Connectedness and path connectedness. Homotopy and homotopy equivalence. Fundamental group. Arun, the son of two math teachers, has always had a strong affinity for math, and sees his teaching as a means to evangelize the relevance of calculus. And this happens outside of the classroom, such as participating in math events.

Those who just did the coursework have more difficulty in research because they didn't develop those creative skills. By teaching this required freshman class, Arun sees firsthand the variety of experience that his students bring to their first year at Berkeley. It's his particular challenge to take that uneven group and bring them to a common understanding by the end of the semester.