138-16-FALL-WEEK2-Sept.19,23-H2.pdf

138-16-FALL-WEEK2-Sept.19,23-H2.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #2 (SEPTEMBER 19 AND 23) Part 2. More on “Solving Problems” and “Thinking logically” . - Some examples and exercises: 1) Exercise 5, page 44. Let 𝐴𝐴 = { 𝑥𝑥 ∈ ℝ | 𝑥𝑥 > 1} , 𝐵𝐵 = { 𝑥𝑥 ∈ ℝ | 𝑥𝑥 ≤ 15} and 𝐶𝐶 = { 𝑥𝑥 ∈ ℝ | | 𝑥𝑥 | < 5} . a) Find 𝐴𝐴 ∩ 𝐵𝐵 . b) Find 𝐵𝐵 ∩ 𝐶𝐶 . c) Find 𝐴𝐴 − 𝐵𝐵 . 2) Exercise 6, page 45 (a,b,c). List the elements in each of the following sets: a) { 𝑥𝑥 ∈ ℝ | 3 𝑥𝑥 + 4 = 0} . b) { 𝑥𝑥 ∈ ℝ | 𝑥𝑥 2 = 9} . c) { 𝑥𝑥 ∈ ℝ | 𝑥𝑥 2 + 1 = 0} . 3) Exercise 9, page 45. Let 𝐴𝐴 = �𝑥𝑥 , 𝑦𝑦 , { 𝑥𝑥 , 𝑦𝑦 } . a) Is { 𝑥𝑥 , 𝑦𝑦 } ⊆ 𝐴𝐴 ? b) Is { 𝑥𝑥 , 𝑦𝑦 } ∈ 𝐴𝐴 ? c) Is { 𝑦𝑦 } ⊆ 𝐴𝐴 ? d) Is { 𝑦𝑦 } ∈ 𝐴𝐴 ? - Variables and Predicates: Statements containing a phrase like “if 𝑥𝑥 and 𝑦𝑦 are any two real numbers such that …” are very common and useful in mathematics. The symbols 𝑥𝑥 and 𝑦𝑦 used in the previous example to represent unspecified elements of a certain set (in our case, unspecified elements of the set ) are usually called “variables”. Sometimes we use subscripts to differentiate several related variables. In a statement like “the product 𝑛𝑛 1 × 𝑛𝑛 2 × 𝑛𝑛 3 of any three consecutive natural numbers 𝑛𝑛 1 , 𝑛𝑛 2 and 𝑛𝑛 3 is …”, the variables 𝑛𝑛 1 , 𝑛𝑛 2 and 𝑛𝑛 3
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