MATH 23B Exemption Exam

Math 23b, "Introduction to Proofs" is a required course for mathematics majors. (It is not required for minors, but it is recommended, as it is a prerequisite for most of the more advanced mathematics courses. Note that Math 23b does not count as an elective towards the minor.) The course introduces students to proof techniques, and emphasizes the writing of mathematics. Math 23b satisfies the University's writing-intensive requirement. It is currently offered every semester. A few students each year have strong enough mathematical backgrounds that they don't need to take Math 23b. We offer the Math 23b exemption exam for them. Passing this exam exempts a student from the Mathematics Department's requirement of Math 23b for the major (and allows a student to enroll in any course requiring Math 23b as a prerequisite). It does not grant credit for Math 23b and it does not satisfy the University's writing-intensive requirement. You should also know that the prerequisite to any course can be waived by permission of the instructor; however, passing a course for which Math 23b is a prerequisite does not exempt you from the Math 23b requirement.

Textbooks that have been used in the recent past for this course are "The Art of Proof" by Matthias Beck and Ross Geoghegan, "A Discrete Transition to Advanced Mathematics" by Bettina Richmond and Thomas Richmond, "Mathematical Proofs: A Transition to Advanced Mathematics, by Gary Chartrand, Albert D. Polimeni, and Ping Zhang", and "Mathematical Thinking: Problem-Solving and Proofs" by John P. D'Angelo and Douglas B. West. However, many other books also cover this material (most books on proofs or discrete mathematics, for example).

The material of COSI 29a is somewhat similar to that of Math 23b, but passing COSI 29a does not exempt you from Math 23b, and many students have taken Math 23b after taking COSI 29a. Students who have taken COSI 29a and done very well in it will have a good chance of passing the Math 23b exemption exam, but average COSI 29a students will probably not pass it. (Computer science majors who have taken Math 23b may be excused from the COSI 29a requirement at the discretion of the computer science department, but this is not automatic.)

The Math 23b exemption is given each year early in the fall semester. It can also be given in the spring semester by request. The exam will be given this term on

Thursday, September 7, from 4-5 pm, 209 Goldsmith

Please contact Ruth Charney ( you wish to take the exam this term.

If you pass the exam, we will need to fill out a form for the registrar that includes your sage ID (8 digits, starting with 2), your class year (20xx), your Brandeis mailbox, and your email address, so please have this information handy.

The exam will ask for proofs or disproofs of various statements that may involve integers, real numbers, sets, functions, equalities, and inequalities. Methods of proof may include direct proofs, indirect proofs, proofs by contradiction, and proofs by induction. Proofs must be reasonably well written, since writing mathematics is an important part of Math 23b. There will be four problems, but some may have more than one part. The problems are not especially difficult; they are designed to test your ability to write proofs rather than your general mathematical ingenuity.

If you have any questions about the exam, contact Ruth Charney x6-3071, email

Here are some sample problems to give you some idea of what might be on the exam. Past exams are not available.

  1. A function f is defined on the positive integers by f(1) = 1 and for n > 1, if n is even then f(n) = 2f(n/2) and if n is odd then f(n) = f(n-1). Prove by induction that f(n) ≤ n for all n ≥ 1.
  2. Prove by contradiction that there do not exist positive integers x and y such that x2 - y2 = 1.
  3. Prove that for every positive real number ε there exists a positive integer N such that for every real number x > N, we have ε2 > 1/x.
  4. A function f: A → B is called injective (or one-to-one) if for all x and y in A, f(x) = f(y) implies x = y. A function g: A → A is called idempotent if for every x in A, g(g(x)) = g(x). A function h: A → A is an identity function if for every x in A, h(x) = x. Prove that an injective idempotent function is an identity function.
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