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 for Math 23b (and enables 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 Math 23b requirement.

The textbook that is currently used for Math 23b is "Mathematical Thinking: Problem-Solving and Proofs", by John P. D'Angelo and Douglas B. West. The first four chapters of this book cover the material that will be on the exemption exam. 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, usually a week after the first day of classes. It will be given on Thursday, September 4, 2008 from 4 to 5 pm in Goldsmith 100.

If you need to take the exam at some other time (not necessarily at the beginning of the school year), this can be arranged, and you should contact me to make arrangements to take it.

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 Ira Gessel, x6-3060, email gessel@brandeis.edu.

Here are some sample problems to give you some idea of what might be on the exam. (Some of the real problems may be a bit harder than these.) Past exams are not available.

1. Prove by induction that for every nonnegative integer n, 2 ⁿ > n.
2. Let x and y be real numbers. Prove by contradiction that if xy <0 then x ≤ 0 or y ≤ 0.
3. A set X of real numbers is called bounded if there exists a real number M such that for all x in X, | x | < M. If X and Y are sets of real numbers, then X+Y denotes the set {x + y | x is in X and y is in Y }. Prove that if X and Y are bounded sets then so is X+Y.
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)) =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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