1. sets and basic notations

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1. sets and basic notations

  1. 1. *A set is a well-defined collection of distinctobjects. The objects in a set are called theelements, or members of the set. If anelement a belongs to set A, then we write .Otherwise, we write .
  2. 2. *Example 1: Which of the following collections aresets?*The past presidents of the country*Good players who have played for the national teamExample 2:A = {1,2,3,4,5}B = {x/x is an even number}M = {5,10,15,…,75}Q is a set of the presidents of the Republic ofthe Philippines
  3. 3. *Roster or Tabular MethodThe roster method involves listing down the elementsof the set where a comma serves to distinguish eachdistinct element of the set and enclosing them inbraces.Examples:A = {1,3,5,…}B = {Monday, Tuesday, Wednesday, Thursday, Friday,Saturday, Sunday}P = {circles, triangles, squares, rectangles, spheres,pentagons}R = {2,4,6,…,20}
  4. 4. *Rule Method or Defining Property Method or SetBuilder NotationThe rule method involves describing a set in terms ofits characteristic or property where only thosedescribed or agreeing to that specific property areconsidered elements of the set. It takes the form,A = {x/”___”}*Examples:A = {x/x is a positive odd number}B = {x/x is a day of the week}C = {x/x is a letter in the English alphabet}D = {x/3 < x < 10}
  5. 5. *The cardinality of a set or the cardinalnumber of a set is the number of elementsin the set A and is denoted by n(A).A set is said to be finite if it is empty or if itconsists of exactly n elements where n is acounting number. Otherwise it is an infiniteset.
  6. 6. **The set of all possible elements underdiscussion is called Universal set and isdenoted by U.*A set with no elements is called an emptyset or null set and is written as Ø or { }.
  7. 7. **Two sets A and B are equivalent if theyhave the same number of elements, that is,n(A) = n(B).*Two sets A and B are equal if and only ifthey contain exactly the same elements.*Two sets are said to be disjoint if they haveno common elements.*Two sets are overlapping or joint sets ifthey have at least one element in common.
  8. 8. **A is a subset of B, denoted by , ifevery element of A also belongs to B.Find all subsets of A = {1,2,3}*If and A ≠ B, then we say that A is aproper subset of B and we write .BABABA
  9. 9. **The power set of A is the set whoseelements are the subsets of A. it is denotedby ρ(A) and the cardinal number is n(ρ(A)) =2n(A).Find the power set of A if A = {5, 10, 15}
  10. 10. **A Venn Diagram (named after Robert Venn)is a geometric representation whichillustrates the relationships between andamong sets. It uses circles usually picturedwithin a rectangle (universal set).UBxAA B U12 34 56
  11. 11. **The union of two sets A and B, denoted byis the set containing all the elementsthat belong to A or B*Example: Find the union of the given pairsof sets:1. A = {4,5,6} ; B = Ø2.BA}6,5,4,3{}3,2,1{
  12. 12. **The intersection of two sets A and B,denoted by , is the set containing theelements that are common to both A and B.Example:Find the intersection of the given sets*A = {4,5,6} B = {4,8,10}*X = {m,s,a} Y = {m,a,t.h.s}*A = {a,f,g} B = {1,2,3,4}BA
  13. 13. **The complement of a set A denoted by A’ isthe set of all elements in the universal set Uthat are not found in A, that isA’ = { .Examples: Find the following sets:*A’ if U = {a, b, c} and A = {c}*B’ where U = {4,6,8,9} and B = {4,8}}/ AxUx
  14. 14. **The difference of two sets A and B,denoted by A – B, is the set of all elementsin B that are not in A.Example: Find A – B.*A = {w,x,y,z} B = {x,z}*A = {a,b,c,d} B = { }*A = {2,3,4,5,6} B = {1,3,5}
  15. 15. **The symmetric difference of two sets A andB denoted by A – B is the setA – B = {x/ and }.*The Cartesian product of A and B denotedby A x B is the set of all ordered pairs (x,y)where and , i.e.A x B = {(x,y)/ and }.BAx BAxAx BxAx Bx

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