MATH 23B Exemption Exam

Math 23b, "Introduction to Proofs" is a required course for mathematics majors. (It is not required for applied math majors or math minors, but it is recommended, as it is a prerequisite for many 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. Math 23b is currently offered every semester. A few students each year have strong enough backgrounds in theoretical mathematics that they don't need to take Math 23b. We offer the Math 23b exemption exam for those students. Passing this exam exempts a student from the Mathematics Department's requirement of Math 23b for the major but it does not satisfy the University's writing-intensive requirement.

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 6, from 5-6 pm, 209 Goldsmith

Please contact Ruth Charney (charney@brandeis.edu) you wish to take the exam this term.

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.

Textbooks that have been used in the recent past for Math 23b are "The Art of Proof" by Matthias Beck and Ross Geoghegan, "A Discrete Transition to Advanced Mathematics" by Bettina Richmond and Thomas Richmond, and "Mathematical Proofs: A Transition to Advanced Mathematics”, by Gary Chartrand, Albert D. Polimeni, and Ping Zhang. However, many other books also cover this material (most books on proofs or discrete mathematics, for example).

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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