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### Exam1

Course: MATH 199, Fall 2009
School: Concordia Chicago
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Word Count: 724

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I Math Exam 199, Spring 08 1. Dene the following terms and expressions: (a) A B A B = (a, b) a A, (b) A function from X to Y . bB , Last First A function from X to Y is a subset f X Y where, for every x X, there is unique element y Y so that (x, y) f . In this case we write f (x) = y. (c) A transitive relation. A transitive relation in X is a subset R X X so that if (a, b) R and (b, c) R then (a, c)...

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I Math Exam 199, Spring 08 1. Dene the following terms and expressions: (a) A B A B = (a, b) a A, (b) A function from X to Y . bB , Last First A function from X to Y is a subset f X Y where, for every x X, there is unique element y Y so that (x, y) f . In this case we write f (x) = y. (c) A transitive relation. A transitive relation in X is a subset R X X so that if (a, b) R and (b, c) R then (a, c) R. (d) Field of rational numbers Q. (Dene elements as equivalence classes. What are the operations?) We dene a relation on (p, q) p, q Z and q = 0 by letting (p, q) (a, b) if aq = bp. This is an equivalence relation and Q is the set equivalence classes. We denote the class containing (p, q) by p . Now dene operations + and by q p a bp + aq + = q b qb and p a pa = . q b qb (e) Ring of integers mod n, Zn . (Dene elements as equivalence classes. What are the operations?) We dene a relation n on Z by letting a n b if n divides (b a). This is an equivalence relation and Zn is the set equivalence classes. We denote the class containing a by a. Now dene operations + and by a+b=a+b and ab=ab Math 297 University of Chicago Spring 2008 2. Mark true or false. Justify you answers briey. (a) For functions f : X Y and g : Y Z, if g f is surjective then g is surjective. This is true. Proof: Pick z Z; we have to show z = f (y) for some y Y . Since g f is surjective, there is x X so that (g f )(x) = z. Therefore, letting y = f (x) Y , we have z = f (y). (b) For functions f : X Y and g : Y Z, if g f is injective then g is injective. This is false. Counter example: Let X = {1}, Y = {1, 1} and Z = {1}. Let f (x) = x and g(y) = y 2. Then (g f )(x) = x is injective, but g is not. (c) Every ring has a unit. This is false. For example the ring of even Or integers. R = {0}. (d) The following subset of R is an interval xR | This is false. 1 <1 x2 = x2 > 1 = x > 1 or x < 1. 1 < 1. x2 This set is a disjoint union of two intervals (, 1) (1, ). Page 2 of 4 Please go on to the next page. . . Math 297 University of Chicago Spring 2008 3. Prove the following theorem: Theorem. If a and b are positive real numbers, then there exists a natural number n such that na > b. See the proof in the textbook. Theorem 3.2.1 (Archimedean Property of R). 4. Dene a relation on R as follows: x y if (y x) Z. (a) Show that this is an equivalence relation. x R, x x = 0 Z. Therefore, x x. That is is reexive. Let x y for x, y R. Then (y x) Z. Since every element in Z has an additive inverse, we have (x y) Z. Therefore, y x. That is is symmetric. Let x y and y z for x, y, z R. Then (y x) Z and (z y) Z. Since Z is closed under addition, we have (z y) + (y x) = (z x) Z. Therefore, x z. That is, is transitive. (b) Describe the equivalence classes. Denote the equivalence class conta...

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