138-16-FALL-WEEK7-Oct.31,Nov.4-H2.pdf

138-16-FALL-WEEK7-Oct.31,Nov.4-H2.pdf - UNIVERSITY OF...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #7 (NOVEMBER 4) SOME BASICS ON “NUMBER THEORY”. - A few useful introductory notions. - Definition. We say that a number 𝑚𝑚 ∈ ℝ is a “minimal element” of a set 𝑆𝑆 ⊆ ℝ if and only if the number 𝑚𝑚 satisfies the following two conditions: i) 𝑚𝑚 ∈ 𝑆𝑆 , and ii) ∀𝑥𝑥 ∈ 𝑆𝑆 , 𝑚𝑚 ≤ 𝑥𝑥 . Similarly, 𝑀𝑀 is a “maximal element” of 𝑆𝑆 ⊆ ℝ means that i) 𝑀𝑀 ∈ 𝑆𝑆 , and ii) ∀𝑥𝑥 ∈ 𝑆𝑆 , 𝑥𝑥 ≤ 𝑀𝑀 . Exercise: Prove that ∀𝑆𝑆 ⊆ ℝ , 𝑆𝑆 cannot have two or more distinct minimal (or maximal) elements. Note: If 𝑚𝑚 is the minimal element of 𝑆𝑆 , then we write 𝑚𝑚 = min( 𝑆𝑆 ) . We say that 𝑚𝑚 is the minimum of 𝑆𝑆 . Also, if 𝑀𝑀 is the maximal element of 𝑆𝑆 , then we write 𝑀𝑀 = max( 𝑆𝑆 ) . We say that 𝑀𝑀 is the maximum of 𝑆𝑆 . Exercise: Find min( 𝑆𝑆 ) and max( 𝑆𝑆 ) , if any, for each of the following sets 𝑆𝑆 ⊆ ℝ . a) 𝑆𝑆 = { 𝑥𝑥 ∈ ℤ | ( 𝑥𝑥 − 1) 2 < 6} , b) 𝑆𝑆 = { 𝑥𝑥 ∈ ℝ | ( 𝑥𝑥 − 1) 2 < 6} , c) 𝑆𝑆 = { 𝑥𝑥 ∈ ℤ | 𝑥𝑥 + | 𝑥𝑥 | < 2016} , d) 𝑆𝑆 = �𝑥𝑥 ∈ ℝ | 5𝑥𝑥−3 5𝑥𝑥−7 0 , e) 𝑆𝑆 = �𝑥𝑥 ∈ ℝ | 𝑥𝑥 ( 𝑥𝑥−1 ) ( 𝑥𝑥+2 )( 𝑥𝑥−3 ) = 1 , and f) 𝑆𝑆 = �𝑥𝑥 ∈ ℚ | 2 − 𝑥𝑥 ∈ ℚ� . - Definition. We say that a set 𝑆𝑆 ⊆ ℝ is “finite”, if 𝑆𝑆 = or if 𝑆𝑆 ≠ ∅ but for some 𝑛𝑛 ∈ ℕ + there exists 𝑓𝑓 : 𝑆𝑆 → {1,2,3, … , 𝑛𝑛 } such that
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