138-16-FALL-PS3.pdf - UNIVERSITY OF TORONTO DEPARTMENT OF...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 PROBLEM SET #3. DUE: NOVEMBER 25 (2:10 p.m.). - Please use different pages for each of the assigned problems. - Staple all your papers and clearly write your name and student number in each of the pages. - Show and explain all the steps in each of your solutions. - Marks will be deducted for messy or unclear solutions. - Late submissions will not be accepted. 1. Let 𝑓𝑓 : ℤ ⟶ 𝒫𝒫 ( ) be defined ∀𝑚𝑚 ∈ ℤ by 𝑓𝑓 ( 𝑚𝑚 ) = �𝑛𝑛 ∈ ℤ│ |5 𝑚𝑚 − 33| 99 ≤ 𝑛𝑛 ≤ |3 𝑚𝑚 − 44| 66 . Let 𝐴𝐴 = �𝑚𝑚 ∈ ℤ│𝑓𝑓 ( 𝑚𝑚 ) ≠ ∅� , let 𝐵𝐵 = �𝑚𝑚 ∈ ℤ│ | 𝑚𝑚 | > 20 and let 𝑊𝑊 = 𝒫𝒫 ( ) { } . a) Find the values of min( 𝐴𝐴 ) and max( 𝐴𝐴 ) . b) List all the elements of each of the three sets 𝑓𝑓 (max( 𝐴𝐴 )) , 𝑆𝑆 𝑆𝑆∈𝑓𝑓 ( 𝐵𝐵 ) and 𝑓𝑓 −1 ( 𝑊𝑊 ) .
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