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P Q’ P  Q P, but not Q Q, but not P neither P nor Q

Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad , Glen Whiffen , John Owen, Robert Haese , Sandra Haese and Mark Bruce Haese and Haese Publications, 2004. Section 3E –Numbers in Regions.

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P Q’ P  Q P, but not Q Q, but not P neither P nor Q

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  1. Material Taken From:Mathematicsfor the international student Mathematical Studies SLMal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark BruceHaese and Haese Publications, 2004

  2. Section 3E –Numbers in Regions Example 1)If (3) means that there are 3 elements in the set P ∩ Q, how many elements are there in: • P • Q’ • P  Q • P, but not Q • Q, but not P • neither P nor Q

  3. Example 2)Given n(U) = 30, n(A) = 14, n(B) = 17 and n(A ∩ B) = 6, find: • n(A B) • n(A, but not B)

  4. Example 3)A squash club has 27 members, 19 have black hair, 14 have brown eyes and 11 have both black hair and brown eyes. • Place this information on a Venn diagram. • Hence find the number of members with: • Black hair or brown eyes • Black hair, but not brown eyes

  5. Example 4)A platform diving squad of 25 has 18 members who dive from 10 meters and 17 who dive from 5 meters. Everyone dives. How many dive from both platforms?

  6. Example 5)A city has three football teams A, B and C, in the national league. In the last season, 20% of the city’s population went to see team A play, 24% saw team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play. Using a Venn diagram, find the percentage of the city’s population which: • Saw only team A play • Saw team A or team B play but not team C • Did not see any of the teams play.

  7. Example 6) One year, 37 students took an exam in physics, 48 took an exam in chemistry and 45 took an exam in biology. 15 students sat exams in physics and chemistry, 13 sat exams in chemistry and biology, 7 sat exams in physics and biology and 5 students took exams in all three. • Draw a Venn diagram to represent this information. • Calculate n(PCB) • Calculate n(P ∩ C) • Calculate n(B ∩ C) • How many students took an exam in only one subject?

  8. Example 7) In a group of 125 students who play tennis, volleyball or football, 10 play all three. Twice as many (as all three) play tennis and football only. Three times as many (as all three) play volleyball and football only, and 5 play tennis and volleyball only. If x play tennis only, 2x play volleyball only and 3x play football only, find: • How many play tennis • How many play volleyball • How many play football.

  9. Homework • Chapter 3 Review Sheet • Review Set 3A and 3B

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