138-16-FALL-WEEK6-Oct.21-H1.pdf

138-16-FALL-WEEK6-Oct.21-H1.pdf - UNIVERSITY OF TORONTO...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #6 (OCTOBER 21) Part 1. SOME BASICS ON “FUNCTIONS”. - Definition. A “function 𝑓𝑓 from 𝐴𝐴 to 𝐵𝐵 ” is a relation from 𝐴𝐴 to 𝐵𝐵 that satisfies the property: ∀𝑥𝑥 ∈ 𝐴𝐴 , there is exactly one 𝑦𝑦 ∈ 𝐵𝐵 such that 𝑥𝑥𝑓𝑓𝑦𝑦 . Example: Let 𝐴𝐴 = { 𝑎𝑎 , 𝑏𝑏 , 𝑐𝑐 } and 𝐵𝐵 = {1,2,3,4,5} . The relation 1 = {( 𝑎𝑎 , 4), ( 𝑏𝑏 , 3), ( 𝑐𝑐 , 3)} ⊆ 𝐴𝐴 × 𝐵𝐵 is a function from 𝐴𝐴 to 𝐵𝐵 because it satisfies the required condition but the relation 2 = {( 𝑎𝑎 , 1), ( 𝑏𝑏 , 3)} is not a function from 𝐴𝐴 to 𝐵𝐵 because 𝑐𝑐 ∈ 𝐴𝐴 and there is no 𝑦𝑦 ∈ 𝐵𝐵 such that 𝑐𝑐ℛ 2 𝑦𝑦 . Also, 3 = {( 𝑎𝑎 , 4), ( 𝑏𝑏 , 1), ( 𝑏𝑏 , 3), ( 𝑐𝑐 , 3)} is not a function from 𝐴𝐴 to 𝐵𝐵 because 𝑏𝑏 ∈ 𝐴𝐴 and there are two different 𝑦𝑦 ∈ 𝐵𝐵 ( 𝑦𝑦 = 1 and 𝑦𝑦 = 3 ) for which 𝑏𝑏ℛ 3 𝑦𝑦 . Example: The relation 𝑓𝑓 = {( 𝑎𝑎 , 𝑏𝑏 ) ∈ ℤ × | 𝑏𝑏 = | 𝑎𝑎 |} is a function from to because ∀𝑎𝑎 ∈ ℤ , there is a unique 𝑏𝑏 ∈ ℕ such that 𝑏𝑏 = | 𝑎𝑎 | . Example: The relation 𝑔𝑔 = {( 𝑎𝑎 , 𝑏𝑏 ) ∈ ℕ × | 𝑎𝑎 = 𝑏𝑏 2 } is not a function from to because for some 𝑎𝑎 ∈ ℕ , there are two different 𝑏𝑏 ∈ ℤ such that 𝑎𝑎 = 𝑏𝑏 2 (for example 𝑎𝑎 = 9 , with 𝑏𝑏 = 3 and 𝑏𝑏 = 3 ).
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern