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University of Toronto St. George

Mathematics

MAT137Y1

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Summer

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MAT137Y1a.doc Lecture #1 Tuesday, September 9, 2003 S ETS non-example: A ={paintings that are be}utiful example: A ={natural numbers divisibl} by 4 Notation 4 A 4 is in A 3 A 4 is not in A B ={naturalnumbersdivisible}y7 AB = {aturalnumbersdivisibleby4or}7 union of A and B = 4,7,8,12,14,...} AB = naturalnumbersdivisibleby4and}7 intersection of A and B FACTS A BOUT R EAL N UMBERS N = natural numbers 1,2,3,4}... Z = integers {...,3,2,1,0,1}2,3,... Q = rational numbers{1,2,3,4}... R = real numbers = all this and more (+ irrational numbers) Geometrically -5 -4 -3 -2 -1 0 1 2 3 4 5 The Real Line any point on the line represents a real number Intervals 1,2]={x R :1 x } 1,2)={x R :1< x < } 0,) Ordering (Inequalities) a < b If a, b are real numbers, then exactly one of the folla = bis true: a > b Important Properties If a < b and c > , then ac < bc If a < b and c < , then ac > bc Page 1 of 33 www.notesolution.com MAT137Y1a.doc DistanceAbsolute Values a =distance fromato 0= a,ifa 0 a,ifa< 0 ab = distancebetween a andb Triangle Inequality a+b a + b analogous to triangle theorem from geometry a b c c a +b c Lecture #2 Thursday, September 11, 2003 T RIANGLE NEQUALITY a+b a + b,a,bR Proof (A Proof By Cases) 1) If a,b both 0 Then a +b 0 a+b = a+b = a + b 2) If a > 0,b < 0, a b Then a +b 0 a+b = a+b a + b = ab Because b < 0 , so a +b < a b 3) If Then a +b 0 a+b = a+b = a + b 4) a > 0,b < 0, a < b 5) 6) Page 2 of 33 www.notesolution.com MAT137Y1a.doc R EVIEW OF INEQUALITIES Solve 2 x+ 6 2x + 3) x1 )x4 )x+2 )> 0 2 x 3 2) 2x 2 Where is it = 0? x =1,4,2 3 x 1 1) x 2 x [1,) -2 1 4 x , 3 - - + + 2 (x1 ) (x4 ) - - - + (x+2 ) - + + + Product - + - + x (2,) (4,) Inequalities With Absolute Values x - 0 < 0 < x < it means thax (, ) x xc < c - c c + it means that < x < c+ x c,c+ ) x >5 x > 5 -2 3 8 x < 5 ( 2 )8,( ) C OORDINATE G EOMETRY Rectangular Coordinates y-axis 3 2 1 (3,1) 1 2 3 x-axis Page 3 of 33 www.notesolution.com

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