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Set

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  • 1. SET 1
  • 2. SETS A group of objects A collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics 2
  • 3. SETS Example:i. Set of even numbers less than 10 2,4,6,8ii. Set of vowels : a,e ,i,o,uiii. Colours of rainbow : red, orange, yellow, green, blue, indigo and violet 3
  • 4. SETSCan be defined by * description * using set notation,e.g i. A set of students of WMS ii. 2,3,5,7 4
  • 5. Sets are named with capital letters and the objects are enclosed in bracese.g : Set A is a set of the first seven numbers in multiple of 7 ( description) A = 7,14,21,28,35,42,49 (set notation) 5
  • 6.  The objects are known as elements of the set. set notation /symbol : : is not an element of A = 7,14,21,28,35,42,49 21 is an element of set A 21 A 8 A 6
  • 7. Set also can be represented by a Venn diagramVenn Diagram: an enclosed geometrical figure, i.e circle, triangle, rectanglee.g P = factors of 9 = 1,3,9 P .3 .9 .1 7
  • 8. n ( A ) means the number of elements of set A.e.g i. P = 1,4,9,16 ,25 n(P)=5 ii. Q = AVATAR Q = A, V, T ,R n (Q) = 4 8
  • 9. P .3 P= 3P 3 n( P ) = 3 9
  • 10. Empty Sets : A set which does not have any elementsSymbol : ore.g Set of odd numbers which can be divided by 2. =e.g A = A polygon with two sides 10
  • 11. Equal sets:Consider sets A= 1,2,3,4 and B = 1,4,2,3 . All elements of set A are elements of set B , hence set A is equal to set B. A = Be.g K = 11,12,13,14 and L = 11,14,12 The element 13 is not found in set L, therefore, K and L are not equal sets. K L 11
  • 12. e.g Given that A = prime numbers less than 15 B = 2, 3, 5, 11, 13 , y +2 and A = B, find the value of y.Solution : A = 2, 3, 5, 7, 11, 13 y+2=7 y=5 12
  • 13. SUBSETSet P is a subset of set Q if all the elements of set P are also found in set Q. “ Set P is subset of set Q” and is written as P Q“ set P is not a subset of set Q” is denoted by P Q 13
  • 14. e.g Given that A = even numbers from 2 to 16 B = 4, 16 C = perfect squares less than 20 . State whether each of the following is true or false. a) B A b) B C c) C A d) A B 14
  • 15. Venn Diagram : P Q Q P 15
  • 16. P = 5, 6, 10 and Q = 6,10 . P Q .10 .5 .6 Q P 16
  • 17. Listing the subsets of a set .i. P = 2 Subsets of P : , 2ii. P = 2, 4 Subsets of P : , 2 4 2,4iii. P = 2,4,6 Subsets of P = , 2 4 , 6 2,4 2,6 4,6 , 2,4,6 17
  • 18. # 1 : an empty set is a subset of any set#2 : A set is a subset of itself#3 : number of subsets of set P is given by n ( A) 2 18
  • 19. Universal set.- the set that contains all the elements under discussion and is denoted by the symbol .- any set under discussion is a subset of the universal set. 19
  • 20. 20
  • 21. e.g Given that = x: 4 x 10, x is an integer P = x: x is a multiple of 3a. List all the elements of and P.b. Draw a Venn diagram to represent and P. 21
  • 22. Complement set :The complement , A’ , of set A is a set which contains all elements in the universal set, , which are not elements of set A.e.g = 2,3,4,5,6,7,8 A = 3,5,7,8 A’ = 2, 4,6, 22
  • 23. AA’ 23
  • 24. OPERATIONS ON SETSIntersection of two sets. The intersection of two sets , A and B, is the set of all the common elements of A and B, and is denoted as A B 24
  • 25. Intersection of two setsThe intersection of two sets , A and B, is the set of all the common elements of A and B, and is denoted as A B 25
  • 26. A BA B 26
  • 27. e.g Given P = 3, 4, 5, 7 , Q= 4,5, 7, 8 and R = 6, 8 , 9 . Find the following set; a. P Q b. Q R c. P RSolution: a. 4,5,7 b. 8 c. or 27
  • 28. Relationship between A B and A or B. ( A B ) A and ( A B) B 28
  • 29. P Q RShade the region that represents P Q R’ 29
  • 30. e.g In a class of 40 students, 22 of them play badminton while 24 play football.Among these students,14 students play both badminton and football. a. Draw a Venn diagram to represent the information above b. Find the number of students who play neither badminton nor football. 30
  • 31. Union of sets. - The union of two sets , A and B is a set that consists all the elements found in the sets A and B.- Denoted by A B ( A union B) 31
  • 32. e.g A = 2,4,6,8 and B = 4,8,10 A B = 2,4,6,8,10 n( A B )= 5- A (A B ) and B (A B) 32
  • 33. A computer centre has 120 students. 76 ofthem study Java while 60 of them studyPascal. Given that 35 of them study both.Find the number of students who study Javaor Pascal. 33

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