BACKGROUND 11 for all a, b, c e D, and we say that it is a commutative operation if a x b = b x a for all a, b e D. Finally let's tie functions, equivalence relations, and sets with operations together. Often an equivalence relation is defined by a collection T of bijective functions such that two objects A and B are said to be equivalent (or more frequently, isomorphic) if there is a function / e T with f(A) = B. When the objects A and B are sets with operations (so that we have (A, x) and (B, +), for example), then we will require the function / to satisfy f(a x a!) = f(a) + f(af) for all a, a! e A. Some of this may be bewildering. Don't worry. It is the purpose of this course to provide examples that shed light on these abstractions. The material of calculus beyond the functional concept will play very little role in this course, although the derivative of a polynomial will surface occasionally. A larger role will be played by linear algebra. Many of the fundamental concepts will be reviewed when needed. A basic familiarity with matrix algebra (addition and multiplication) and the elementary properties of the trace and determinant of a matrix will be assumed. Exercises 0.3. Make up an example of a relation that is symmetric and transitive but not reflexive. Make up another that is reflexive and transitive but not symmetric. Make up a third that is reflexive and symmetric but not transitive. Occasionally proofs in the text, and proofs requested of you as exercises, will best be done by Mathematical Induction. Sometimes students come to believe that Mathe- matical Induction is a "silver bullet" which should be used for all proofs. This is far from the case. Rarely is "induction" the appropriate tool in any of the exercises. When it is, you will usually be advised to use it. The Principle of Mathematical Induction. Suppose that {P(0), P(l),...} is a set of mathematical statements indexed by the natural numbers. Suppose further that the following two statements are true: (a) The statement P (0) is true. (b) For any natural number n, the truth of P(n) implies the truth of P(n + 1). Then all of the statements P(0), P(l), ... are true. In fact this is really just a statement about sets of natural numbers: Let S be a set of natural numbers about which the following statements are true: (a) 0 is in S. (b) Ifn is in S, then also n + 1 is in S. Then S is the set N of all natural numbers.
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