For sets A and B

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					Venn Diagram
• A useful way of representing various ___________.

• A collection of ____________________ or different _____________.
• Denoted by ______________ in a Venn diagram.

• The individual ________________ of a set.

Universal Set
• The set ___, from which all the _____________ are derived.
• Denoted by a _________________ box enclosing the _________.

Common Elements
• Elements which belong to more than _____ set.
• Illustrated by using ___________________ circles.
• Example: If S = the set of whole numbers less than 12

                A = the set of even numbers less than 12

                B = the set of numbers less than 12 which are divisible by 3

Subset of a Set
• “A” is a subset of “B” if all the elements in ____ are also in ____.
• Example: If S = { real numbers}

                B = { whole numbers}

                C = { even whole numbers}
Union of Sets
• The union of sets A and B is denoted by ____________.
• It consists of all elements which are in A _____ B.

Intersection of Sets
• The intersection of sets A and B is denoted by ____________.
• It consists of all elements which are in A _____ B.

Principle of Inclusion and Exclusion

•   For sets A and B, the total number of elements in either A or B is the number in
    A plus the number in B minus the number in both A and B.

    n ( A ⋃ B) =

•   It can also be applied to three or more sets.


1. There are 10 students on the volleyball team and 15 on the basketball team.
   When planning a field trip for both teams, the coach arranges transportation
   for 19 students.
      a. Why?
      b. Draw a Venn Diagram
      c. How many students are on both teams?
      d. What might the set S represent?

2. There are 140 students in Grade 12. Of them, 42 take Biology, 71 take
   Chemistry, 40 take Physics, 15 take Biology and Chemistry, 8 take Chemistry
   and Physics, 11 take Biology and Physics, and 2 take all three sciences. How
   many do not take any sciences at all?

3. There were 25 employees at a meeting. Coffee and tea was served. Of the
   employees, 16 of them liked coffee, 12 of them liked tea, and 5 of them liked
   neither coffee nor tea. How many liked both?
1. A survey of television viewers at A Child’s Place Preschool produces the following data:
      (5 marks)
          60% watch Sesame Street.                                                           C
          50% watch Barney & Friends.
          50% watch Doodlebops.
          30% watch Sesame Street and Barney & Friends.
          20% watch Barney & Friends and Doodlebops.                                         20
          30% watch Sesame Street and Doodlebops.
          10% watch all three shows.
      a) Illustrate the situation with a Venn diagram.
      b) What percentage viewed at least one of these programs?
      c) What percentage viewed none of the shows?
      d) What percentage viewed Sesame Street and Barney & Friends but not
      e) What percentage viewed exactly two of these programs.

2.     Ace Electronics, a growing firm, has applied to Doe Insurance Company for a group
       life insurance policy. The insurance company requests the following data concerning
       the 1605 employees of Ace: the number who are married, the number who are over
       40 years of age, and the number who have passed the required physical
       examination. The personnel manager from Ace provides the following data:
                   715 are married.
                   894 are over 40.
                   911 have passed the physical.
                   352 are married and over 40.
                   365 are married and have passed the physical.
                   320 are over 40 and have passed the physical.
                   209 have passed the physical, are married, and are over 40.
       Show that this information is in error. (Illustrate the situation with a Venn
       diagram and record all your calculations to prove the data error.) (2 marks)

3.     Of 1400 students at Tomlintown High, 800 attended the first school dance of the
       year. The music was not good so only 500 attended the next dance. If 300
       attended both dances, how many did not go to either event? (1 mark)
4.   In a recent election poll of 193 people, the following information was collected:

     140 of those polled were professionals; 84 were under 30 years of age; 133 voted
     Conservative in the last election; 56 were professionals under 30; 41 of those under
     30 voted Conservative; 111 professionals voted Conservative; 30 of the
     professionals under 30 voted Conservative.

     Of those polled, how many non-professionals aged 30 or over did not vote
     Conservative? (1 mark)

5.   In a group of 100 families it was found that 83 subscribe to Time, 41 subscribe to
     Newsweek, and 32 subscribe to both. Find the number of families that: (4 marks)
     a) Draw a Venn diagram to illustrate this situation.
     b) Subscribe to neither,
     c) Subscribe to Time only,
     d) Subscribe to Newsweek only.

6.   In a group of 120 families it was found that 68 subscribe to Time, 47 subscribe to
     Newsweek, and 36 subscribe to Newsweek only. Find the number that: (4 marks)
     a) Draw a Venn diagram to illustrate this situation.
     b) Subscribe to Time only,
     c) Subscribe to neither,
     d) Subscribe to both.

7.   Among 70 freshman students a survey showed that: (3 marks)

           23 were taking physics.
           25 were taking biology.
           22 were taking chemistry.
           6 were taking physics and biology.
           7 were taking biology and chemistry.
           8 were taking chemistry and physics.
           2 were taking all three sciences.

     a) Draw a Venn diagram to illustrate this situation.
     b) How many of the freshmen were taking none of the three sciences?
     c) How many were taking just one of the three sciences?

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