# hw1 - a/b Show that the following operations of...

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Math 113 Homework 1, due 1/26/2012 at the beginning of class Policy: if you worked with other people on this assignment, write their names on the front of your homework. Remember that you must write up your solutions independently. 1. (a) Let f : X Y and g : Y Z . Show that if g f : X Z is onto, then g is onto. Show that if g f is one-to-one, then f is one-to-one. (b) Show that f : X Y is bijective (i.e., one-to-one and onto) if and only if there exists g : Y X with g f = id X and f g = id Y . (Optional: what happens if you drop one of the two conditions on g ?) 2. Fraleigh, section 0, exercises 29–34. 3. In this exercise we will construct Q starting from Z . Let S = { ( a, b ) | a, b Z , b n = 0 } . De±ne a relation on S by ( a, b ) ( c, d ) ⇐⇒ ad = bc. (a) Show that is an equivalence relation. (b) Let Q be the set of equivalence classes, and denote the equivalence class of ( a, b ) by [ a, b ] (ordinarily we would denote this by
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Unformatted text preview: a/b ). Show that the following operations of “addition” and “multiplication” on Q are well de±ned: [ a, b ] + [ c, d ] = [ ad + bc, bd ] , [ a, b ] [ c, d ] = [ ac, bd ] . 4. Show that every positive integer n has a binary expansion, i.e. can be expressed as a sum of distinct powers of 2. For example, 2010 = 2 10 + 2 9 + 2 8 + 2 7 + 2 6 + 2 4 + 2 3 + 2 1 . (Hint: by the division theorem you can write n = 2 q + r with r ∈ { , 1 } ; use induction.) Extra credit: show that the binary expansion of a given positive integer is unique. 5. Fraleigh, section 2, exercises 31, 32, 34. 6. Fraleigh, section 3, exercises 3, 4, 7. 7. How challenging did you ±nd this assignment? How long did it take?...
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