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

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

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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 ).

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• Winter '15
• Math, DOM, Inverse function

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