138-16-FALL-WEEK7-Oct.31-H1.pdf

138-16-FALL-WEEK7-Oct.31-H1.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #7 (OCTOBER 31) More on learning and working with new concepts and definitions and proving properties. - Topic: An “axiomatic” way to describe a “numerical” set. Exercise. Let 𝑊𝑊 denote a set that satisfies each of the following five properties: (P1) ∃ℴ ∈ 𝑊𝑊 , (P2) ∃𝑠𝑠 : 𝑊𝑊 → 𝑊𝑊 , (P3) ∀𝑚𝑚 ∈ 𝑊𝑊 , 𝑠𝑠 ( 𝑚𝑚 ) ≠ ℴ , (P4) the function 𝑠𝑠 is injective, and (P5) ∀𝐴𝐴 ⊆ 𝑊𝑊 , ( ℴ ∈ 𝐴𝐴 ) ∧ �∀𝑘𝑘 ∈ 𝑊𝑊 , ( 𝑘𝑘 ∈ 𝐴𝐴 ⇒ 𝑠𝑠 ( 𝑘𝑘 ) ∈ 𝐴𝐴 ) �� ⇒ ( 𝐴𝐴 = 𝑊𝑊 ) . Give a proof for each of the following propositions: a) 𝑠𝑠 ( ) ≠ ℴ , b) ∃𝐴𝐴 ⊆ 𝑊𝑊 , (| 𝐴𝐴 | = 3) ( 𝐴𝐴 ∩ { } = ) , c) ∀𝑚𝑚 ∈ 𝑊𝑊 , 𝑠𝑠 ( 𝑚𝑚 ) ≠ 𝑚𝑚 , d) ∀𝑚𝑚 ∈ 𝑊𝑊 , 𝑠𝑠 ( 𝑠𝑠 ( 𝑚𝑚 )) ≠ 𝑠𝑠 ( 𝑚𝑚 ) , e) if 𝑊𝑊 + = 𝑊𝑊 − { } , then s ( 𝑊𝑊 ) = 𝑊𝑊 + , and f) if g : 𝑊𝑊 + → 𝑊𝑊 is defined by, ∀𝑚𝑚 ∈ 𝑊𝑊 + , ∀𝑛𝑛 ∈ 𝑊𝑊 , 𝑔𝑔 ( 𝑚𝑚 ) = 𝑛𝑛 ⇔ 𝑚𝑚 = 𝑠𝑠 ( 𝑛𝑛 ) , then 𝑔𝑔 is a bijection. Exercise. Let 𝑓𝑓 : 𝑊𝑊 2 → 𝑊𝑊 be defined by, ∀𝑚𝑚 , 𝑛𝑛 ∈ 𝑊𝑊 , 𝑓𝑓 ( 𝑚𝑚 , 𝑛𝑛 ) = 𝑛𝑛 if 𝑚𝑚 = 𝑠𝑠 ( 𝑓𝑓 ( 𝑘𝑘 , 𝑛𝑛 )) if 𝑚𝑚 = 𝑠𝑠 ( 𝑘𝑘 ) . a) Compute 𝑓𝑓 ( , ) , 𝑓𝑓�ℴ , 𝑠𝑠 ( ) , 𝑓𝑓 ( 𝑠𝑠 ( ), ) , 𝑓𝑓�𝑠𝑠 ( ), 𝑠𝑠 ( ) and 𝑓𝑓 �𝑓𝑓�𝑠𝑠 ( ), 𝑠𝑠 ( ) , 𝑠𝑠 �𝑓𝑓�𝑠𝑠 ( ), 𝑠𝑠 ( ) ��� . b) Give a proof for each of the following propositions: b1) ∀𝑛𝑛 ∈ 𝑊𝑊 , 𝑓𝑓 ( 𝑠𝑠 ( ), 𝑛𝑛 ) = 𝑠𝑠 ( 𝑛𝑛 ) , b2) ∀𝑚𝑚 , 𝑛𝑛 ∈ 𝑊𝑊 , ( 𝑓𝑓 ( 𝑚𝑚 , 𝑛𝑛 ) = )
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