Math 301, Homework 1
…rst draft sue on Thursday October 8
1. Prove that for a function f : S ! T is one-to-one if and only if for all
subsets C1 ; C2 in S we have
f (C1 \ C2 ) = f (C1 ) \ f (C2 ) :
2. Suppose that h : S ! T and k : T ! S are two functions
Math 301, Solutions for Homework 2
1. A librarian decides to make a catalogue of all the books in his library that do not contain the name of the book in the text (except for the title pages of course!). He names this catalogue "Complete list of all books
Math 301, Homework 1
Solutions 1. Prove that for a function f : S ! T is one-to-one if and only if for all subsets C1 ; C2 in S we have f (C1 \ C2 ) = f (C1 ) \ f (C2 ) : Solution 1 Suppose f is 1-2. Take two sets C1 ; C2 in S: Let y 2 f (C1 \ C2 ) : Then
Take Home Exam Solutions
1. For each of the following, either nd an example satisfying the given conditions or give not long but complete explanation (a short proof quoting some known fact or theorem) of why there is or why there is no such example. (a) A
Math 301 Midterm. Solutions
1. (40 points) In your answers you should refer to denitions or theorems/lemmas. (a) Is there a bounded increasing sequence in R which is not Cauchy? If yes give an example, otherwise prove that there is no such example. (b) Co
Math 301, Homework 7
rst draft is due on Thursday January 8 1. For each positive a 2 R; Let Ia =
a 3 ; 3a
:
e e (a) Let U be the collection U = fIa : 0 < a < 3g. Find all integers k so e is an open cover for Ak = (k; k + 3) : that U e (b) For which of the
Math 301, Homework 6
rst draft is due on Thursday December 25 1. Find all accumulation points of the following sets in R2 with the usual metric. o n 1 1 1 (a) ; n + m : n; m 2 N : n+m (b) Q Q (c) D (0; 1) r f(0; 0)g 2. For a set A metric space (M; d) and
Math 301, Homework 5
rst draft is due on Thursday November 13. do not forget to bring the nal version of Homework 3 1. Give the answer with short explanations. The sets below are all considered in R. i. Decide if they are bounded from below/above by nding
Math 301, Homework 4
rst draft is due on Thursday November 6. do not forget to bring the nal version of Homework 2 For problems below do not use any convergence theorem, use only denitions; i.e. give direct proofs by an N argument. All sequences are in R:
Math 301, Homework 3
rst draft is due on Thursday October 23. do not forget to bring the nal version of Homework 1 a b b with a; b a x y if yx
1. Let K be the set of all 2 real. Consider the map z = x + iy
2 matrices of the form : C ! K dened as
(z ) =
(a
Math 301, Homework 2
…rst draft is due on Thursday October 16.
1. A librarian decides to make a catalogue of all the books in his library
that do not contain the name of the book in the text (except for the title
pages of course!). He names this catalogue