1.1. Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers?
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1 1.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS Prepring for 2A.6.K, 2A.7.I Intervl Nottion nd Set Nottion Essentil Question When is it convenient to use set-uilder nottion to represent set of numers? A collection of ojects is clled set. You cn use rces { } to represent set y listing its memers or y using set-uilder nottion to define the set in terms of the properties of its memers. For instnce, the set of the numers 1, 2, nd 3 cn e denoted s {1, 2, 3} List the memers of the set in rces. nd the set of ll odd whole numers cn e denoted s { is whole numer nd is odd} Set-uilder nottion which is red The set of ll rel numers such tht is whole numer nd is odd. If ll of the memers of set A re lso memers of set B, then set A is suset of set B. For instnce, if set A = {, } nd set B = {,, c, d}, then set A is suset of set B. Writing Susets in Set Nottion ANALYZING MATHEMATICAL RELATIONSHIPS To e proficient in mth, you need to connect nd communicte mthemticl ides. Work with prtner. Write ll the nonempty susets of ech set.. {4, 5}. {c, d} c. {2, 4, 6} d. {e, f, g, h} Writing Susets in Set Nottion Work with prtner. Write ech given suset of the rel numers in set-uilder nottion. Descrie ech set-suset reltionship mong these sets.. the integers. the whole numers c. the nturl numers d. the rtionl numers e. the irrtionl numers f. the positive integers Writing Susets in Set Nottion Work with prtner. Write ech indicted set of numers using either rces to list its memers or set-uilder nottion. Eplin your choice of nottion.. the whole numers 50 through 54. the rel numers 0 through 4 c. the prime whole numers d. the integers 00 through 100 Communicte Your Answer 4. When is it convenient to use set-uilder nottion to represent set of numers? 5. Wht re some reltionships etween susets of the rel numers? Section 1.1 Intervl Nottion nd Set Nottions 3
2 1.1 Lesson Wht You Will Lern Core Voculry set, p. 4 suset, p. 4 endpoints, p. 4 ounded intervl, p. 4 unounded intervl, p. 5 set-uilder nottion, p. 6 Represent intervls using intervl nottion. Represent intervls using set-uilder nottion. Using Intervl Nottion In mthemtics, collection of ojects is clled set. You cn use rces { } to represent set y listing its memers or elements. For instnce, the set {1, 2, 3} A set with three memers contins the three numers 1, 2, nd 3. Mny sets re lso descried in words, such s the set of rel numers. If ll the memers of set A re lso memers of set B, then set A is suset of set B. The set of nturl numers {1, 2, 3, 4,...} is suset of the set of rel numers. The digrm shows severl importnt susets of the rel numers. UNDERSTANDING MATHEMATICAL TERMS The symols represent susets of the rel numers. R: Rel numers Q: Rtionl numers Z: Integers W: Whole numers N: Nturl numers Rel Numers (R) Rtionl Numers (Q) Integers (Z) Whole Numers (W) Nturl Numers (N) Irrtionl Numers Mny susets of the rel numers cn e represented s intervls on the rel numer line. Core Concept Bounded Intervls on the Rel Numer Line Let nd e two rel numers such tht <. Then nd re the endpoints of four different ounded intervls on the rel numer line, s shown elow. A rcket or closed circle indictes tht the endpoint is included in the intervl nd prenthesis or open circle indictes tht the endpoint is not included in the intervl. Inequlity Intervl Nottion Grph [, ] < < (, ) < [, ) < (, ] The length of ny ounded intervl, [, ], (, ), [, ), or (, ], is the distnce etween its endpoints:. Any ounded intervl hs fi nite length. An intervl tht does not hve finite length is clled unounded or infi nite. 4 Chpter 1 Liner Functions
3 Core Concept Unounded Intervls on the Rel Numer Line Let nd e rel numers. Ech intervl on the rel numer line shown elow is clled n unounded intervl. Inequlity Intervl Nottion Grph [, ) > (, ) (, ] < (, ) (, ) The symols (infi nity) nd (negtive infi nity) re used to represent the unoundedness of intervls such s [7, ) nd (, 7]. Becuse these symols do not represent rel numers, they re lwys enclosed y prenthesis. Writing Intervl Nottion Write ech intervl in intervl nottion.. 3. > c. 5 3 d. 5 3 SOLUTION. The grph of 3 is the ounded intervl [, 3].. The grph of > is the unounded intervl (, ). c. The grph represents ll the rel numers etween 3 nd 4, including the endpoint 3. This is the ounded intervl [ 3, 4). d. The grph represents ll the rel numers less thn or equl to 3. This is the unounded intervl (, 3]. Monitoring Progress Help in English nd Spnish t BigIdesMth.com Write the intervl in intervl nottion < < Section 1.1 Intervl Nottion nd Set Nottion 5
4 Using Set-Builder Nottion Another wy to represent intervls is to write them in set-uilder nottion. Core Concept Set-Builder Nottion Set-uilder nottion uses symols to define set in terms of the properties of the memers of the set. Set-uilder nottion { < } Words the set of ll rel numers such tht is less thn Set-uilder Nottion Grph { or > } { } Using Set-Builder Nottion Sketch the grph of ech set of numers.. { 2 < 5}. { 0 or > 4} SOLUTION. The rel numers in the set stisfy oth > 2 nd 5.. The rel numers in the set stisfy either 0 or > 4. 6 UNDERSTANDING MATHEMATICAL TERMS The symol denotes memership in set. The epression Z mens tht is memer (or element) of the set of integers. Writing Set-Builder Nottion Write the set of numers in set-uilder nottion.. the set of ll integers greter thn 5. (, ) or (, ) SOLUTION. is greter thn 5 nd is n integer. { > 5 nd Z}. cn e ny rel numer ecept. { } Monitoring Progress Help in English nd Spnish t BigIdesMth.com Sketch the grph of the set of numers. 4. { 6 < } 5. { 0 or 10} Write the set of numers in set-uilder nottion. 6. (, ] or (1, ) 7. the set of ll integers ecept 6 Chpter 1 Liner Functions
5 1.1 Eercises Dynmic Solutions ville t BigIdesMth.com Voculry nd Core Concept Check 1. COMPLETE THE SENTENCE Two rel numers nd re the of four different intervls on the rel numer line. 2. WHICH ONE DOESN T BELONG? The grph of which set of numers does not elong with the other three? Eplin. ( 3, 5] > 3 nd 5 { 3 < 5} the set of ll integers greter thn 3 nd less thn or equl to 5 Monitoring Progress nd Modeling with Mthemtics In Eercises 3 6, use rces to list the elements of the set. 3. the set of whole numers less thn the set of odd whole numers less thn the set of integers greter thn the set of integers less thn 8 In Eercises 76, write the intervl in intervl nottion. (See Emple 1.) 7. 3 < < < < the rel numers from 0 through the rel numers etween 110 nd 220 In Eercises 170, sketch the grph of the set of numers. (See Emple 2.) 17. { 3 < < 12} 18. { 0 15} 19. { < 5 or > 10} 20. { 4} In Eercises 218, write the set of numers in set-uilder nottion. (See Emple 3.) 21. [ 5, 16) 22. (22, 98] 23. (, ] or [4, ) 24. (, 5] or [14, ) 25. the set of ll integers less thn the set of ll rel numers greter thn 19 nd less thn the set of ll rel numers ecept the set of ll whole numers ecept ERROR ANALYSIS Descrie nd correct the error in rewriting the intervl (, 8] in set-uilder nottion. { < 8} 30. ERROR ANALYSIS Descrie nd correct the error in rewriting the intervl [ 7, 24) in set-uilder nottion. { 7 or < 24} Section 1.1 Intervl Nottion nd Set Nottion 7
6 31. MODELING WITH MATHEMATICS The elevtion reltive to se level in the United Sttes rnges from 82 feet in Deth Vlley, Cliforni to 20,320 feet on Mount McKinley, Alsk. Write the rnge of elevtions in intervl nottion nd in set-uilder nottion. 32. MODELING WITH MATHEMATICS The the min floor of n uditorium rnges from 6 feet elow the stge to 8 feet ove the stge. The floor of the lcony rnges from 26 to 37 feet ove the stge. Write the rnge of the floor levels reltive to the stge in intervl nottion nd in set-uilder nottion. 33. MAKING AN ARGUMENT Your friend sys tht it is impossile to write the set { 30 nd 60, nd W} in intervl nottion. Is your friend correct? Eplin. 35. NUMBER SENSE Write ech set using rces to list the elements, in intervl nottion, nd in set-uilder nottion. If not possile, eplin why.. the set of even whole numers. the set of rel numers less thn c. the set of rel numers 10 or more units from THOUGHT PROVOKING Eplin how you cn dd the sme numer to ech memer of the set of whole numers to produce nother importnt suset of the rel numers. 37. MATHEMATICAL CONNECTIONS You re mrking rectngulr pintll zone tht must e 34 meters wide nd hve perimeter of t lest 140 meters ut not more thn 260 meters. Find the intervl for the length of the rectngulr pintll zone. 34. HOW DO YOU SEE IT? The grphs show the legl driving speeds (in miles per hour) on two different rodwys. A Legl speeds (miles per hour) B Legl speeds (miles per hour). Write the legl driving speeds shown in Grph A in intervl nottion, in set-uilder nottion, nd in words.. Write the legl driving speeds shown in Grph B in intervl nottion, in set-uilder nottion, nd in words. c. One of the rodwys is stte highwy nd the other is residentil street. Which grph represents ech rodwy? 38. MATHEMATICAL CONNECTIONS You hve 20 gllons of roof coting to pply to the roof of moile home tht is 16 feet wide. Twenty gllons covers 760 to 1000 squre feet. Find the intervl for the length tht you will cover efore you need to uy more roof coting. 16 ft Mintining Mthemticl Proficiency Reviewing wht you lerned in previous grdes nd lessons Complete the tle of vlues for the function f. Then grph the function. (Skills Review Hndook) 39. f() = f() = f() f() 41. f() = f() = f() f() 8 Chpter 1 Liner Functions
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