Boundedness of sets.pptx

Boundedness of sets, supremum and infimum

1. Boundedness of – { SETS } – Course Title : Real Analysis-1 Course Code MTH-211 Presenting by : Arafat Hossain ID : 12004059 Session : 2019-2020 Dpartment of Mathematics Directed by : MD. Abdullah Al Mahbub (Associate Professor) Dpartment of Mathematics Comilla University

2. Upper bound & Lower bound Upper bound of a set: Let S be any subset of the set R of real numbers. If there exists a real number u such that x ≤u ,  x  S , then u is called an upper bound of the set S . Lower bound of a set : Let S be any subset of the set R of real numbers. If there exists a real number l such that Is l ≤ x ,  x  S , then l is called a lower bound of the set S . Example : Let S = {…-3, -2, -1} Here -1 is an upper bound of S . Example : Let S = {1,2,3,4 ….} Here lower bound of S = 1

3. Bounded & Unbounded sets Bounded set : Let S be any subset of the set R of real numbers. If there exists two real numbers u and I such that, l  x  u then the set S is called bounded . Example : Let S = {1 , 3, 5, 7} Here 1 is a lower bound and 7 is an upper bound Unbounded set : Let S is a set. If there exist no real number I and u such that. I≤x ≤ u hold,  x  S , then S is called an unbounded set. Example : The set R of real numbers is an unbounded set. because it is nether bounded above nor bounded below.

4. Supremum & Infimum Supremum :The least of all the upper bounds of a set is called its supremum or the least upper bound. Example : Let S = {1 , 2. 3. 4} Here 4 is an upper bound of S Supremum of S = 4 . supS =4] Infimum : The greatest of all the lower bounds of a set is called its infimum or the greatest lower bound. Example: Let S = (1, 2, 3, 4) Here ,1 is a lower bound of S .. Infimum of S = 1 . infS =1]

5. EXERCISE Find the supremum and infimum of the set S = {x : 3x² - 10x + 3 <0} Solution : Given S = { x : 3x² - 10x + 3 < 0} Now 3x 2 -10x+3<0 ⇒ 3x²-9x-x+3<0 ⇒ 3x(x-3) -1(x-3) < ⇒ (3x - 1) (x-3) <0 This is true when 3x-1>0 and x-3<0 ⇒ x > 1/3 and x<3 ⇒ 1/3 < x < 3 Or , ( ii) 3x-1 <0 and x-3>0 ⇒ x< and x>3 When x <1/3 then 3x-1 <0 and x-3<0 When x>3 then 3x-1>0 and x-3>0 Thus , for x < 1/3 or x > 3 we have (3x - 1)(x-3) > 0 S =(x : 3x² - 10x +3<0) ⇒ S = {x: 1/3 < x < 3} sup S = 3 inf S = 1/3 ( Ans )

6. Thanks All