138-16-FALL-WEEK9-Nov.14,18-H1.pdf

138-16-FALL-WEEK9-Nov.14,18-H1.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #9 (NOVEMBER 14 TO18) Part 1. Some basics on “Modular Arithmetic”. - Topic: Some important previous results. - Property. If 𝑛𝑛 is any positive integer number, then the relation “ 𝑛𝑛 ” defined on the set by the condition ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , 𝑎𝑎 ≡ 𝑛𝑛 𝑏𝑏 ⇔ 𝑛𝑛 |( 𝑏𝑏 − 𝑎𝑎 ) is an equivalence relation on the set of integer numbers . - Property. Some important and useful properties of “ 𝑛𝑛 ”: i) ∀𝑎𝑎 ∈ ℤ , 𝑎𝑎 ≡ 𝑛𝑛 𝑎𝑎 , ii) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ( 𝑎𝑎 ≡ 𝑛𝑛 𝑏𝑏 ) ( 𝑏𝑏 ≡ 𝑛𝑛 𝑎𝑎 ) , iii) ∀𝑎𝑎 , 𝑏𝑏 , 𝑐𝑐 ∈ ℤ , ( 𝑎𝑎 ≡ 𝑛𝑛 𝑏𝑏 ) ( 𝑏𝑏 ≡ 𝑛𝑛 𝑐𝑐 ) � ⇒ ( 𝑎𝑎 ≡ 𝑛𝑛 𝑐𝑐 ) , iv) ∀𝑎𝑎 , 𝑏𝑏 , 𝑐𝑐 , 𝑑𝑑 ∈ ℤ , ( 𝑎𝑎 ≡ 𝑛𝑛 𝑏𝑏 ) ( 𝑐𝑐 ≡ 𝑛𝑛 𝑑𝑑 ) � ⇒ �� ( 𝑎𝑎 ± 𝑐𝑐 ) 𝑛𝑛 ( 𝑏𝑏 ± 𝑑𝑑 ) � ∧ ( 𝑎𝑎𝑐𝑐 ≡ 𝑛𝑛 𝑏𝑏𝑑𝑑 ) , and v) ( / 𝑛𝑛 ) = {[0], [1], [2], … , [ 𝑛𝑛 − 1]} . Exercises: 1) Give a proof for part (v) of the previous property. 2) Give a proof for each of the following “identities”: a) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ( 𝑎𝑎 + 𝑏𝑏 ) 4 4 ( 𝑎𝑎 2 + 𝑏𝑏 2 ) 2 , b) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , ( 𝑎𝑎 + 𝑏𝑏 ) 5 5 ( 𝑎𝑎 5 + 𝑏𝑏 5 ) , c) ∀𝑎𝑎 , 𝑏𝑏 ∈ ℤ , 2( 𝑎𝑎 + 𝑏𝑏 ) 8 4 2( 𝑎𝑎 8 + 𝑏𝑏 8 ) , and d) ∀𝑎𝑎 ∈ ℤ , (3 2 𝑎𝑎 ) 6 + (1 + 𝑎𝑎 )(1 + 𝑎𝑎 5 ) � ≡ 5 0 .
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