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Venn Diagrams

Venn Diagrams. EQ: How do I use a Venn diagram to represent different sets of numbers and to solve problems?. B. A. U. A. B. U. A. B. U. Disjoint Sets. U. A B =. Disjoint and Exhaustive. U = {x|-3<x<8, x } A and B are subsets of U, A={x|x<5} and B = {x|x 5}.

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Venn Diagrams

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  1. Venn Diagrams EQ: How do I use a Venn diagram to represent different sets of numbers and to solve problems?

  2. B A U

  3. A B U

  4. A B U

  5. Disjoint Sets U AB =

  6. Disjoint and Exhaustive U = {x|-3<x<8, x } A and B are subsets of U, A={x|x<5} and B = {x|x 5}

  7. U={1,2,3,4,5,6,7,8}A = 1,3,6,8 B = {2,3,4,5,8} A B U

  8. U={1,2,3,4,5,6,7,8}A = {1,3,6,7,8} B = {3,6,8} A B U

  9. A B U Four regions on a Venn Diagram • In A, but not in B • In B, but not in A • In both A and B • Neither in A nor B

  10. Q P (7) (11) (3) (4) U Example 1 • Given the Venn Diagram below, how many elements are there in • P • P U Q • Q’ • P, but not Q • Q, but not P • neither P nor Q?

  11. Y X (8) (3) (6) (2) U YOU DO: • Give the number of elements in: • X’ • X Y • X U Y • X, but not Y • Y, but not X • Neither X nor Y

  12. B A 6 + c = 17 ; c = 11 11 Example 2: 8 + 6 + 11 + d = 30 25 + d = 30 ; d = 5 5 • Given n(U) = 30, n(A) = 14, n(B) = 17 and n(A B) = 6 find: • n(A U B) • n(A, but not B) 25 8 (a) (c) • b = 6 • a + b = 14 • b + c = 17 • a + b + c + d = 30 6 (b) a + 6 = 14 a = 8 8 (d) U

  13. B A (a) (c) (b) (d) U YOU DO: • Given n(U) = 26, n(A) = 11, n(B) = 12 and n(A B) = 8, find: • n(A U B) • n(B, but not A) • n(A’) 15 4 15 b = 8 a + b = 11; a = 3 b + c = 12; c = 4 a + b + c + d = 26 3 + 8 + 4 + d = 26; d = 11

  14. Black Brown (a) (c) (b) (d) U Now, the real thing… • 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 • Find the number of members with: • Black hair or brown eyes • Black hair, but not brown eyes d = 5 a + b + c + d = 27 a + b = 19 b + c = 14 a = 8 c = 3 b = 11

  15. Long Brown (a) (c) (b) (d) U YOU DO: Pele has 14 cavies as pets. Five have long hair and 8 are brown. Two are both brown and have long hair. • Place this information on a Venn diagram • Find the number of cavies that: • Are short haired • Have short hair and are brown • Have short hair and are not brown c + d = 9 c = 6 d = 3 a + b + c + d = 14 a + b = 5 b + c = 8 b = 2 d = 3 a = 3 c = 6

  16. 10 m 4 m (a) (c) (b) (d) U A little bit different… • A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 4 m. How many dive from both platforms? • a + b + c + d = 25 • a + b = 18 • b + c = 17 • 18 + c + 0 = 25 • c = 7 • Therefore b + 7 = 17 and b = 10

  17. Now for the real real thing… • A city has three newspapers A, B, and C. Of the adult population, 1% read none of these newspapers, 36% read A, 40% read B, 52% read C, 8% read A and B, 11% read B and C, 13% read A and C and 3% read all three papers. What percentage of the adult population read: • Newspaper A only • Newspaper B or Newspaper C • Newspaper A or B but not C A B d d g g e a = 3; a + d = 8; a + b = 11; a + c = 13 Therefore, d = 5, b = 8, and c = 10 g + 3 + 5 + 10 = 36; therefore g = 36 e + 3 + 5 + 8 = 40; therefore e = 24 f + 3 + 8 + 10 = 32; f = 31 and h = 1 a a c c b b b b b f f f C U U U

  18. HW #4 • 4a – pg 78 #1; #4; pg 79 #7, pg 80 #9 • 4b – pg 81, 82 #11 - 14

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