Applications of Sets

Goals:

• Continue to apply problem solving techniques to solve problems
• Set, Define Venn Diagram
• Use Venn Diagrams to solve set related problems

Background Terminology

• A set is a collection of objects
• Objects in a set are called elements.
• What is wrong with the set c, in example 2 on page 43?
• The Universal Set U is a set that contains everything.
• One way to represent a set is with a Venn Diagram.
  • You draw a rectangle to represent the Universal Set.
  • You draw a circle for each set you wish to represent
  • Your circles should probably intersect with each other.
  • We will mostly deal with two and three set Venn diagrams.
  • 
  • 
  • We do this so we can think about the set without thinking about all of the elements.
  • If an element is in set A, it is within the circle representing A.
  • 
  • If an element is in set A or B it is inside of one of the two circles (or both)
  • 
  • If an element is in set A and B, it is in the small area in both circles.
  • 
  • Example:

    	 
    If  set A is the people who own cats
    and set B is the people who own dogs
    
    The people who own cats and dogs is the area labeled II 
    The people who own cats or dogs are the areas labeled I, II, and III
    
    What does area IV represent?
    	 

  • We represent the number of things in a region as n(region name)
  • For example, the number of people who own dogs is n(I) + n(II)

Applications of Sets

• 
• Example 2: (Problem 2, Page 80)

  Use the two set Venn Diagram
      Let A be the set of students who study in the library
      Let B be the set of students who study in the student lounge
      Known:
       n(U) = 160
       n(I)+n(II) = 79
       n(II)+n(III) = 65
       n(II) = 43
      What do we want to know?
         A) n(I)
         B) n(III)
         C) n(IV)
      How will we do this?
         n(I) = n(I)+n(II) - n(II) = 79 - 43 = 36
         n(III) = n(III) + n(III) - n(III) = 65 - 43 = 22
         n(IV) = n(U) - (n(I) + n(II) + n(III)) = 160 - (79+22)
               = 160 - 101 = 59
      

• Example 3, #8 page 81

  Use the three set Venn Diagram above
  What do we know?
      n(U) = 65
      Let A be the set of resorts that have refrigerators
      Let B be the set of resorts that have laundry service
      Let C be the set of resorts that have child care
      n(A) = n(I) + n(II) + n(IV) + n(V) = 34
      n(B) = n(II) + n(III) + n(V) + n(VI) = 30 
      n(C) = n(IV) + n(V) + n(VI) + n(VII) = 37 
      n(II) + n(V)  = 15
      n(IV) + n(V) = 17
      n(V) + n(VI) = 19
      n(V) = 7
  What do we want to know?
     n(I)
     n(I) + n(III) + n(VII)
     n(U) - n(VIII)
     n(II) + n(IV) + n(VI)
     n(VIII)
  Plan
     Solve a simpler problem first
     Find n(II)
        n(II) = n(II) + n(V) - n(V)
              =      15      -  7
  	    = 8
     Find n(IV) 
        n(IV) = n(IV) + n(V) - n(V)
              = 17 - 7 
  	    =  10
     n(I) = n(A) - n(II) - n(V) - n(IV)
          = 34 - 8 - 10 - 7
  	= 26 - 10  - 7
  	= 16 - 7
  	= 9
        n(I) = 
      

Homework

Please do 1-11 odds, pages 80-92