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Sets, Vectors & Functions

Sets, Vectors & Functions. IGCSE Chapter 8. Note 1 : Sets. A ∩ B is green. A. B. ∩ - intersection U – Union - ‘is a subset of’. A. B. ∩. B. A. ∩. A B. Note 1 : Sets. X. a. C - ‘is a member of’ ‘belongs to’ b є X ξ – ‘universal set’ ξ = { a,b,c,d,e }

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Sets, Vectors & Functions

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  1. Sets, Vectors & Functions IGCSE Chapter 8

  2. Note 1: Sets A ∩ B is green A B ∩ - intersection U – Union - ‘is a subset of’ A B ∩ B A ∩ A B

  3. Note 1: Sets X a C - ‘is a member of’ ‘belongs to’ bєX ξ – ‘universal set’ ξ = {a,b,c,d,e} - ‘complement of’ b c ξ d a c b e ξ B A ‘ A A U A’ = ξ

  4. Note 1: Sets A a n(A) - ’the number of elements in set A’ A = {x : x is an integer, 2 ≤ x ≤ 9} Reads: A is the set of elements x such that x is an integer and 2 ≤ x ≤ 9 b n(A) = 3 c The set A is {2,3,4,5,6,7,8,9} ∩ Ø or { } - ‘empty set’ Ø A for any set A

  5. Note 1: Sets ξ T R e.g. In the venn diagram 8 5 11 ξ = {students in year 10} 9 13 15 R = {Yr 10 students who play Rugby} 12 17 V = {Yr 10 students who play Volleyball} V T = {Yr 10 students who play Tennis} a.) How many students play Rugby? b.) How many students do not play Volleyball? c.) How many students play Rugby and Tennis? d.) How many students in total? e.) How many play only 1 of these 3 sports? 40 41 14 90 31 IGCSE Ex 1 Pg 242-243

  6. e.g. If X = {1,2,3,..10} ξ Y = {1,3,5,…19} Y X 13 1 4 Z = {x : x is an integer, 5≤x≤11} 15 3 17 Find: 2 5 7 19 10 {1,3,5,7,9} 9 a.) X ∩ Y b.) Y ∩ Z c.) X ∩ Z d.) n(X U Y) e.) n(Z) f.) X’ U Y’ 11 6 8 {5,7,9,11} {5,6,7,8,9,10} Z 15 7 Ø or { } State whether true or false: False a.) 7 εX ∩ Y b.) {5,7,9,11} Z c.) Z X U Y ∩ True IGCSE Ex 2 Pg 244 ∩ True

  7. e.g. Draw and shade this diagram to show the following sets: a.) X∩ Y b.) (X U Y)’ c.) X’ ∩ Y ξ X Y IGCSE Ex 3 Pg 245-246

  8. e.g. Logical Problems If A = { sheep } B = { sheep dogs } C = { ‘intelligent animals’ } D = { good pet } Express the following in set language: a.) No sheep are ‘intelligent’ animals b.) All sheep dogs make good pets c.) Some sheep make good pets A ∩ C = Ø ∩ B D A ∩ D = Ø Interpret the following statements a.) B C b.) B U C = D c.) AεD ∩ All sheep dogs are intelligent animals All sheep dogs and all intelligent animals make good pets Sheep do not make good pets

  9. e.g. Logical Problems Of 27 students in the class, 18 play chess, 15 play piano and 7 do both. How many do neither? C P 27 students = 11 + 7 + 8 + X 7 11 8 27 – 11 – 7 – 8 = X X = 1 1 There is only 1 student who does not play either piano or chess

  10. e.g. Logical Problems The math results from the Hockey Team show that all 16 players passed at least 2 subjects, 8 passed at least 3 subjects and 5 students passed 4 subjects or more ξ How many passed exactly 2 subjects? What fraction passed exactly 3 subjects? 3 2 3 8 8 5 4 3/16 IGCSE Ex 4 Pg 247-249

  11. Note 2: Vectors A vector is a quantity that has both magnitude and direction Vectors can be added using scale drawings. Head to Tail method: a a b a + b Therefore: a + b = b + a b * Notice that this is the same result as b + a

  12. Note 2: Vectors A scalar is a quantity that has magnitude but no direction. e.g. ordinary numbers, & quantities like temperature, mass & volume. We can multiply a vector by a scalar. e.g.x multiplied by 2 gives 2x x x x x x x -3x 2x e.g.x multiplied by -3 gives -3x * The negative sign reverses the direction of the vector

  13. Note 2: Vectors The result of a – b is the same as a + -b e.g.Find 3a – b a a a a a a a a b b b b 3a – b e.g.Find -4a + 2b -4a + 2b

  14. Note 2: Vectors d Starting from Eeach time, find vectors for the following: c e.g. Find: 2c 3c + d -c + d -d c – 4d -2c – 4d EG EA EC EF EL EK C A B E F D G J H I M L K

  15. Note 2: Vectors d The result of a – b is the same as a + -b c e.g. Find: DE DG HJ GB BE GC CG = 2c C A B = c – 2d F E D = c– 2d = 2c + 3d = -c – d G H = 4c + 3d = -4c – 3d J I

  16. Note 2: Vectors Write each vector in terms of a, b and c e.g. • AB DE DC HD FE BE EA DG A C 2a a = 2a a B D = -a-c b c c = -4a F = 2a+c b = b +a a a G H E = -2b ( c ) = -2b + 2a = 2b – 2a

  17. Note 2: Vectors Write each vector in terms of a, bonly e.g. DE DC HD A C 2a a a B D = -3a+ 2b b c c = -4a F = 4a– 2b b a a G H E IGCSE Ex 5 Pg 251-252

  18. Note 2: Vectors OA = a and OB = b e.g. Using the figure, express each of the following vectors in terms of a and/or b AP = 3OA OB = 2BQ NP = QN = 3a a.) AP b.) AB c.) OQ d.) PO e.) PQ f.) PN g.) ON h.) AN i.) BP j.) QA = -a + b = 3b = -4a IGCSE Ex 6 Pg 253-255 = -4a + 3b = ½PQ = -2a + 3/2b Q = 4a + PN = 2a + 3/2b b = 3a + PN = a + 3/2b N b B = -b + 4a b A P O = -3b + a a a a a

  19. Note 3: Column Vectors The vector AB can be written as a column vector ( ) 2 Horizontal Component (Movement in x direction) AB = 3 Vertical Component (Movement in y direction) F C B ( ) ( ) ( ) -2 5 0 -3 3 0 E A H G D

  20. Note 4: Vectors coming soon……

  21. Note 5: Vectors

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