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# Review Sheet for MATH 300 at UMass

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

UMASS AMHERST MATH 300 FALL 708 F HAJIR EXAM 1 REVIEW The material you should know for Exam 1 is as follows everything that appears in HWl7 HW27 HW3 and HW47 including the reading speci ed in those homework assignments The exam will have three parts 1 De ntions7 2 Short Answer7 and 3 Problems The problems will usually but not always require you to write a cogent7 concise and correct proof Some of the problems will be statements that were already proved in class or are taken directly from homework But at least some of the problems will require you to prove a statement that has not been presented to you before7 or to nd a counterexample to a statement For the de nitions7 it is important to be extremely precise For instance ifl ask you de ne what it means for f X a Y to be surjective7 the response f is surjective means that for all y 6 Y7 there exists z E X such that f y receives full credit7 and everyting in Y gets hit by somebody in X77 receives only partial credit because gets hit by77 is not suf ciently precise Here begineth the sample exam The points will be distributed approximately as follows 25 De nitions7 25 Short Answer7 and 50 Problems You may wish to take this exam in a quiet room without notes under a time constraint or not7 this is just a suggestion it may be a good suggestion for some students and not so good for others The actual exam will be somewhat similar7 but not identicalll7 to this one This practice exam is LONGER than the actual exam Sample Exam 1 1 DEFINITIONS A set is A set X is nite means that A bijection from X to Y is If f X a Y and g Y a Z are maps7 then the composite map 9 o f has source and target and is de ned by A set X is equal to a set Y means that A set X is a subset of a set Y means that The intersection of X and Y is de ned by The power set 73X is A partition of a set X is We say that two statements P and Q are equivalent if The direct or Cartesian product of X and Y is the set X gtlt Y A map f X a Y is invertible if 1 2 MATH 300 EXAM 1 REVIEW 2 SHORT ANSWER The converse of 7P i 7Q is The negation of P Q is 7P Q Let R be the statement Whenever it rains my car gets wet State the negation of R Determine whether 7P V 7Q gt 7P Q is a tautology Write down two sets X and Y say which are equivalent but not equal Construct the truth table for P i Q i Q i P and determine whether it is equivalent to Q i P Give a bijection 01 a 01 other than the identity map For each statement below Indicate whether it is True or False le is equal to Y then X Q Y and Y Q X If f X a Y is surjective and g X a Y is injective then 9 o f is bijective If X and Y are equivalent sets then X is nite if and only if Y is nite If X is an in nite set then every subset of X is equivalent to X If X is a set then Q X leQYandYQZthenXQZ If X is nite then 73X is also nite The set N 1 23 is equivalent to the set y E Zly 21984 If X is a non empty subset of a set Y then XY X is a partition of Y If f X a Y is surjective and g Y a Z is surjective then 9 o f is surjective 3 PROBLEMS 1 Prove that if X is a set then there is an injection f X gt 73X 2 Suppose ABC are sets a Show that C A U B Q C A b Give explicity three sets A B C such that C A U B is a proper subset of C A 3 Suppose f X a Y and g Y a Z are given maps and let h gof a Prove that if h is surjective then so is g b Specify an example of X Y Z fg as above so that g is surjective but f isn7t Drawing a clear and detailed picture will suf ce 4 Let f X a Y be a map Suppose there exist functions 9 Y a X and h Y a X such that gfx z for all z E X and fhy y for all y E Y i Show that g h ii Show that g is bijective

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