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Set Theory - DOC

VIEWS: 24 PAGES: 19

  • pg 1
									                                                     Set Theory


                                                         Part A

Group 1: Fill in the blank with the best answer to make each statement correct.

1. A(n) ____________ is a collection of objects.

2. The objects in the collection are called ____________ or ____________ of the set.

3. ____________ letters are used to designate sets.

4. Draw the symbol used to designate sets:

         a. is a member of a set: ____________

         b. is not a member of a set: ____________

5. Braces { } are used to display the ____________ of a set.

6. One way to name a set is to list the names of the elements between braces such as {1, 2, 3} or
{violin, piano, drum}, this is referred to as ____________ method or ____________.

7. Another way to name a set is to state a rule which describes the elements of the set such as
{even numbers} or { x x is an odd number}, this is referred to as the ____________ method or

___________________.

8. D = {2, 4, 6, 8} is read “______________________________________________________.”

9. E = { x x is a one digit even number} is read “ ____________________________________.”

10. 3  A is read (or means) “____________________________________________________.”

11. 5  B is read (or means) “____________________________________________________.”

12. Because of the pattern established, the notation {1, 2, 3, 4, 5, …} denotes the set of all
____________ numbers.

13. Represent as a set the letters in the word Mississippi: ________________________.

14. If N = {1, 2, 3, 4, 5, …}, then 0 _______ N.

15. If C = {2, 4, 6, 8, 10, …}, then 7  C , but 12 _______ C.


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                                                     Set Theory



Group 2: Write each of the following sets in words:

1. {1, 3, 5, 7, 9}                                                11. F = { x x is a whole number}

2. {a, b, c, d, e}                                                12. G = { x x is a province of Canada}

3. {Buffalo, Chicago, Boston, Cleveland}                          13. H = { x x is a student at Holland Central}

4. {3, 6, 9}                                                      14. 5  A

5. {5, 10, 15, 20, … 100}                                         15. 20  B

6. {2, 4, 6, 8, …}                                                16. Mary  {John, Sue, Mary}

7. B = {5, 6, 7, 8, 9}                                            17. 17  {5, 10, 15, 20}

8. C = {4, 8, 12, 16, 20, …}                                      18. 9  {1, 2, 3, 4, 5, …}

9. D = { x x is a one-digit number}                               19. 24  { x x is a two digit-number}

10. E = { x x is an even two-digit number}                        20. 0  { x x is a natural number}


Group 3: Use the roster method and tabulate the elements of the set that is described. When
impractical to list all the elements, use the ellipses.

1. The set of the months of the year that begin with the letter J.

2. The set of natural numbers greater than 7 and less than 12.

3. The set of one-digit odd numbers.

4. The set of even numbers greater than 2 and less than 5.

5. The set of all whole numbers.

6. The set of the months of the year which have exactly 30 days.

7. The set of all natural numbers between 10 and 1000.

8. The set of odd numbers between 1 and 50 which are exactly divisible by 3.




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                                                     Set Theory

9. The set of all U.S. Presidents since John F. Kennedy.

10. The set of all proper fractions whose numerators and denominators are chosen from the
numbers 1, 2, 3, 4.

11. The set of all proper fractions whose numerator is the number 1.

12. The set of letters in the word little.

13. The set of all natural numbers exactly divisible by 5.


Group 4: Use the rule method and describe a rule for each set illustrated below.

1. {Erie, Ontario, Michigan, Superior, Huron}

2. {3, 4, 5, 6, 7}

3. {Tuesday, Thursday}

4. {a, e, i, o, u}

5. {10, 11, 12, 13, … 99}

6. {2, 4, 6, 8, 10, …}

7. {21, 23, 25, 27, 29}


Group 5: List the elements of each set.

1. { x x is an even number between 6 and 13}

2. { x x is the team who won last year’s World Series}

3. { x x is a planet in our solar system}

4. { x x is an odd number greater than 10}

5. { x x is a three-digit number}

6. { x x is a whole number but not a natural number}




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                                                     Set Theory


7. { x x is an even number greater than 3 and less than 95}

8. { x x is a major league sports team representing Buffalo}



Group 6: Which of the following sets are well defined? Explain.

1. { x x is a state of the U.S.}

2. { x x is a baseball team in the National League}

3. { x x is an honest man}

4. { x x is an even number between 5 and 11}

5. { x x is a small natural number}

6. { x x is the largest continent on Earth}

7. { x x is a small body of water}

8. { x x is the largest city in New York State}



Group 7: Write using the proper symbol.

1. Seven is a member of set B.

2. Five is not a member of set C.

3. Dan is a member of set M.

4. z is not a member of set H.

5. r is a member of set F.

6. Thirteen is not a member of the set of one-digit numbers.

7. Buffalo is a member of the set of teams in the N.F.L.

8. Mars is a member of the set of planets in our solar system.


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                                                     Set Theory



Group 8: True or False?

         Given:              A = {2, 4, 6, 8, 10, 12}
                             B = {2, 4, 8, 10}
                             C = {4, 10, 12}
                             D = {2, 10}
                             E = {all natural numbers}

1. 6  B                                      6. 4  C                  11. 12  U

2. 8  A                                      7. 2  A                  12. 5  D

3. 8  C                                      8. 12  C                 13. 0  U

4. 6  D                                      9. 5  U                  14. 4  D

5. 10  A                                     10. 6  C                 15. 10  C



                                                          Part B
Group 9: Fill in the blank with the best answer to make each statement correct.

1. For a set to be well defined, we must be able to say whether or not any given object is a(n)
____________ of the set.

2. If every member of one set is also a member of a second set, we say that the first set is a(n)
____________ of the second set.

3. A(n) ____________ set contains a limited number of elements.

4. A set which contains no elements is referred to as the ____________ or ____________ set.

5. Sets which contain the same number of elements are called ________________.

6. ____________ are sets which contain exactly the same elements.

7. A(n) ____________ set contains an unlimited number of elements.

8. The following symbol,  , is used to designate the ____________ set.

9. Every set is a(n) ____________ of itself.




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                                                     Set Theory



10. The empty set is considered a(n) ____________ of every set.


Group 10: Identify each set as finite or infinite.

1. {letters in our alphabet}

2. {possible combinations of the letters in our alphabet}

3. {red blood cells in your body}

4. {whole numbers}

5. {fractions between 0 and 1}

6. {stars in our galaxy}

7. {all two-digit numbers}

8. {fractions whose numerator is 1 and whose denominator is a whole number}

9. {fractions whose numerator is a whole number and whose denominator is 1}

10. {grains of sand at Crystal Beach}

11. {odd natural numbers}

12. {students at Holland Central}

13. {numbers divisible by 10}

                                                     1
14. {“numerals” which have a value of                  }
                                                     2

                                                    1
15. {“numbers” which have a value of                  }
                                                    2


Group 11: Identify each set as finite, infinite or empty; then represent or illustrate each set
using the roster method.

1. { x x is an even number}




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                                                     Set Theory


2. { x x is a three-digit number}

3. { x x is an even number or x is an odd number}

4. { x x is an even number and x is an odd number}

5. { x x is an even number greater than 11}

6. { x x is a whole number greater than 1000}

7. { x x is a whole number less than 1000}

8. { x x is a whole number which is not a natural number}

9. { x x is a natural number which is not a whole number}

10. { x x is a whole number greater than 39 and less than 40}

11. { x x is a whole number less than 39 or greater than 40}

                                                           1
12. { x x is a numeral which has a value of                  }
                                                           3


Group 12: Answer the question below about equal sets and equivalent sets.

1. Which of the following sets are pairs of equal sets?

         A = {3, 4, 7, 9}                                         E = {2, 3, 4, 6, 7}
         B = {4, 3, 9, 5}                                         F = {7, 4, 9, 3}
         C = {7, 4, 3, 2, 6}                                      G = {4, 7, 3, 8, 6}
         D = {7, 4, 2, 5, 6}                                      H = {3, 4, 5, 9}

2. Tell whether or not there is a one-to-one correspondence between the two sets.

         a. {5, 6, 9, 4, 8} and {1, 3, 5, 6, 7}
         b. {a, b, c, d} and {x, y, z}
                               1 1 1 1
         c. {5, 6, 7, 8} and { , , , }
                               8 7 6 5
         d. {odd numbers between 50 and 60} and {odd numbers between 60 and 70}
         e. {all even natural numbers} and {all natural numbers}


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                                                     Set Theory

3. Which of the following sets are pairs of equivalent sets?

         A = {0, 1, 2, 4, 5}                                      E = {1, 7, 6, 2}
         B = {8}                                                  F = {3, 13, 23}
         C = {0, 1, 3, 6, 9, 10}                                  G = {31, 32, 33, 34, 35}
         D = {12, 13, 14, 15}                                     H = {3, 2, 9, 6, 12, 8, 1}

4. Given:          A = { x x is a letter of the word “follow”}
                   B = { x x is a letter of the word “wolf”}
                   C = { x x is a letter of the word “flow”}

         a. Are sets A and B equal?
         b. Are sets A and C equal?
         c. Are sets B and C equal?

5. Given {B, A, T} and {T, A, B}, are these two sets equivalent? Are these two sets equal?

6. Given {even natural numbers between 1 and 10} and {odd natural numbers between 1 and
10}, are these two sets equivalent? Are these two sets equal?

7. Describe two finite sets that are equivalent sets but not equal sets.

8. Describe two finite sets that are not equivalent sets.

9. Describe two infinite sets that are equivalent sets.


Group 13: Answer the following questions about subsets.

1. Write each of the following statements in words:
       a. G  H
       b. R  S
       c. {1, 2, 3}  {0, 1, 2, 3, 4, 5}
       d. {5}  {natural numbers}


2. Write in symbols:
       a. y is a subset of Z
       b. D is a subset of E
       c. {3, 4} is a subset of {3, 4, 5}
       d. {6, 7, 8} is a subset of {natural numbers}
       e. {July} is a subset of {months of the year}




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                                                     Set Theory

3. If A = {1, 3, 5, 6, 8, 9}, and B = {1, 5, 6, 8}, and C = {1, 5, 6, 7, 8}
       a. Is B a subset of A?
       b. Is C a subset of A?
       c. Is B a subset of C?

4. True or False?

         a. {8, 9, 10}  {1, 2, 3, 4, 5, …}
         b. {8, 9}  {2, 4, 6, 8, …}
         c.   {1, 3, 5, 7, …}
         d. {5, 10, 15}  {5, 10, 15}
         e. {whole numbers}  {natural numbers}
         f. {r, s, t}  {t, s, r}
         g.   {0}
         h. {odd numbers}  {even numbers}
         i. {1, 2, 3}  
         j. B  B
         k. {Washington, Jefferson, Lincoln, Roosevelt}  {all Presidents of the U.S.A.}
         l. {all odd whole numbers}  {all natural numbers}
         m. {all natural numbers exactly divisible by 4}  {all even natural numbers}
         n. { }  {whole numbers}
         o. {5, 8, 7}  {7, 8, 9, 10}

5. Given:      R = {all odd whole numbers}, S = {all whole numbers}, G = {10, 20, 30, 40},
H = {5, 15, 25, 35, 45}, and L = {all multiples of 5}. Are the statements true or false?

         a. H  C                                                 d. G  R
         b. G  L                                                 e. H  L
         c. H  G                                                 f. R  S

6. Write all the possible subsets for each of the following sets.

         a. {3}


         b. {salt, pepper}


         c. {0, 1, 2}


         d. {all even natural numbers less than 10}




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                                                     Set Theory


                                                         Part C
Group 14: Fill in the blank with the best answer to make each statement correct.

1. A       B is the ____________ of A and B.

2. A       B is the ____________ of A and B.

3. The union of A and B, A               B, is the set that contains all the ____________ in A or B.

4. If x  A         B, the x  of A _______ B.

5. The intersection of A and B, A                B, is the set which contains all the elements of __________
A and B.

6. If x  A         B, then x  A _______ x  B.

7. If x  A and x  B, then x  A                 B but x ________ of A             B.

8. If A and B have no common elements, then A                          B = ____________.

9. The complement of A s written ______________.

10. The ____________ is the name given to the collection of all objects under discussion.

11. A’ is the set off all elements in the universal set which are ____________ elements of A.

12. Every object in the universal set is either in A or in ____________.


Group 15: Intersection of sets.

1. If A = {1, 3, 5} and B = {0, 1, 2, 3}, find A                  B.

2. If R = {6, 7, 8} and S = {3, 4, 5, 6, 7}, find R                S.

3. If C = {5, 10, 15, 20} and D = {5, 6, 7, 8, 9, 10}, find C                  D.

4. If X = {1, 3,4, 6} and Y = {1, 2, 3, 4}, then find:
       a. X Y
       b. Y X
       c. Is it true for all sets X and Y, that X   Y=Y                      X?




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                                                     Set Theory

5. If H = {0, 3, 5, 6, 9}, K = {3, 4, 9}, and L = {1, 8}, find:
       a. H K
       b. H      L

6. If M = { 1, 3, 5, 7, 9} and N =  , find M                     N.


Group 16: Union of sets.

1. If A = {0, 2, 4} and B = {1, 2, 3, 4}, find A                   B.

2. If C = {1, 2, 3, 4, 5} and D = {2, 4, 6, 8, 10}, find C                    D.

3. If R = {1, 3, 5} and S = {0, 2, 4, 6}, find R                  S.

4. If X = {1, 2, 3} and Y = {6, 7, 8}, find:
       a. X Y
       b. Y X
       c. What can be said about X        Y and Y                      X?

5. If M = {1, 3, 5, 7, 9} and N =  , find M                  N.


Group 17: Determine each union or intersection.

1. {5, 6, 19, 35}          {8, 12, 19}                                 7. {a, b, c, d, e}      {e, c, b, a, d}


2. {1, 5, 6, 9, 2}         {3, 4, 8, 7, 10}                            8. {3, 4, 8, 7, 10}       {1, 5, 6, 9, 2}


3. {4, 8, 7, 3}        {14, 3, 7, 12}                                  9. {1, 2, 3, 4, 5, 6}      {5, 2, 4}


4. {6, 4, 19, 35}          {19, 12, 8}                                 10. {4, 3, 1}        {3, 5, 1, 2, 4}


5. {a, b, c, d, e}        {b, e, a, c, d}                              11. {x, y, z}        {x, y, z}


6. {7, 3, 12, 14}          {4, 8, 7, 3}                                12. {x, y, z}        {x, y, z}




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                                                     Set Theory


Group 18: Determine each union or intersection.

1. If A = {1, 2, 3, 4}, B = {2, 4, 6}, C = {5, 6}, D = {1, 3}, and E = { }, find:

         a.   A     D                                             f. D A
         b.   A     B                                             g. (B  C)        D
         c.   B     C                                             h. (B  C)        D
         d.   B     D                                             i. B E
         e.   A     D                                             j. A  E

2. If A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}, find:

         a. A       B                                             i. B C
         b. A        C                                            j. A  A
         c. B       C                                             k. (A B)   C
         d. B       B                                             l. A (B   C)
         e. (A       B)       C                                   m. A   (B  C)
         f. A       (B       C)                                   n. A (B   C)
         g. A       B                                             o. (A  B)  (A            C)
         h. A        C



Group 19: Complement of a set.

1. If U = {1, 3, 5, 7, 9, 11, 13, 15, 17}, find the complement of each set below:

         a.   A = {1, 3, 5}
         b.   B = {7, 15}
         c.   C={}
         d.   D = {1, 3, 5, 7, 9, 11, 13, 15, 17}
         e.   E = {13}


2. If U = {m, a, t, h, i, s, o, k}, find the complement of each set below:

         a.   F = {m, a, t, h}                                    e. J = {m, a, t, h, i, s, o, k}
         b.   G = {i, s}                                          f. K = {m, i, o}
         c.   H = {o, k}                                          g. L = {h, o, t}
         d.   I={}




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                                                     Set Theory



Group 20: Union, intersection and complement.

1. Given the following sets, find:
       U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
       A = {1, 2, 5, 7, 9}
       B = {2, 3, 4, 6, 8}
       C = {5, 6, 9, 10}

         a. A B                                                   d. ( A B )
         b. A B                                                   e. (A  B)         C
         c. ( A B )                                               f. (A     B)      C


Group 21: Union, intersection, complement, and disjoint sets.

1. If U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 1, 2, 3, 4, 5}, B = {2, 3, 4, 5}, C = {4, 5, 6, 7} and
D = {6, 7, 8, 9}, find:

         a. A       B                                             j. D   U
         b. B       C                                             k. B
         c. C       D                                             l. C
         d. D       A                                             m. ( A B )
         e. B       D
         f. B       C                                             n. ( B    D)
         g. A                                                    o. (U     )
         h. B        
         i. C       U


2. Which of the above sets, if any, are “disjoint sets”?


3. If U = {a, b, c, d, e, … z}, A = {a, b, c, d, e}, B = {e, f, g, h, i}, C = {i, j, k, l, m}, find:

         a.   A      B                                            e. A     
         b.   B      C                                            f. B     U
         c.   C      
         d.          A




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                                                     Set Theory



Group 22: Union, intersection, and complement.

If U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, D = {0, 1, 2}, E = {2, 3, 4, 9}, and F = {0, 1, 2, 3, 4}, find:

         1.   D     D                                             10. ( D    F)
         2.   F     E                                             11.   D   F
         3.   E     E                                             12.   D F
         4.   D     F                                             13.   D F
         5.   D     D                                             14.   D E
         6.   F     F                                             15.   D   F
         7. D E                                                   16. D     E
         8. F E                                                   17. ( D E)
         9. ( D E )                                               18. ( E    D)

Group 23: Union and intersection.

1. If A = {1, 4, 8, 9}, B = {2, 3, 5, 7}, C = {1, 2, 3, 8} and D = {1, 2, 3, 4, 5, 7}, find:

         a. (A      C)      D                                     g. (A     C)     B
         b. (A      D)      C                                     h. (A     B)     C
         c. (C      D)      A                                     i. (A     B)    (A C)
         d. (B      C)      D                                     j. (A     B)    (A C)
         e. (A      B)      C                                     k. (A     B)     (B C)
         f. (A      C)      B                                     l. (A     C)    (B D)

2. If A = {1, 2, 3, 5, 6}, B = {2, 4, 5, 8} and C = {1, 3, 5, 7}, then:

         a. Does A         B = B A?
         b. Does A         B = B A?
         c. Does (A        B) C = A (B C)?
         d. Does (A         B) C = A (B C)?
         e. Does A        (B C) = (A B)  (A C)?
         f. Does A        (B C) = (A B)  (A C)?




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                                                     Set Theory


                                                         Part D
Group 24: Examine the Venn diagram and write the set requested by listing its elements.

                                                                  1.   U
                                                                  2.   A
                                                                  3.   B
                                                                  4.   A    B
                                                                  5.   A    B
                                                                  6.   A
                                                                  7.   B
                                                                  8.   (A   B)
                                                                  9. ( A B )




Group 25: Examine the Venn diagram and write the set requested by listing its elements.

1. U
2. A
3. B
4. C
5. A B
6. A B
7. B C
8. B C
9. A
10. B
11. A (B C)
12. (A B) C
13. (A B) C
14. (A B) C
15. (A B) (A C)
16. ( A B )
17. ( A C )
18. ( A C )
19. Express {5, 6} in terms of A, B and C.




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                                                     Set Theory


Group 26: Answer each group of questions below.

1. U = {natural numbers}, A = {10, 11, 12, 13, 14, 15}, B = {13, 14, 15, 16, 17}.
       a. List the elements of A B.
       b. Use a Venn diagram to illustrate A B.




2. U = {natural numbers}, A = {even natural numbers less than 20}, and B = {natural numbers
less than 20 that are divisible by 4}.
        a. List the elements of set A and set B.
        b. List the elements of A B.
        c. Use a Venn diagram to illustrate A B.




3. U = {natural numbers}, A = {10, 11, 12, 13}, and B = {11, 12, 13, 14}.
       a. List the elements in A B.
       b. Use a Venn diagram to illustrate A B.




4. U = {all odd natural numbers less than 20}, A = {1, 5, 7, 11, 15}, and B = {3, 5, 9, 11, 19}.
Draw a Venn diagram to illustrate all of the following:
       a. A B         b. A B         c. A            d. B




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                                                     Set Theory

5. U = {0, 1, 2, 3, … 12}, L = {1, 2, 5, 9, 10, 11}, M = {0, 2, 4, 6, 8, 10}, and N = {2, 5, 7, 10,
12}. Draw a Venn diagram to illustrate all of the following.

         a. L M
         b. M N
         c. L N
         d. N
         e. L
         f. N      M
         g. N      L
         h. M
         i. M      L
         j. N      M


Group 27: Answer the questions below.

1. Draw a Venn diagram showing the relationships among the given sets. List the members of
set F.
       a. E = {5, 6, 7}; E F = {3, 4, 5, 6, 7}; E F = {5, 7}
       b. E = {8, 9, 11}; E F = {8, 9, 10, 11, 12}; E F = {8}




2. If set A has 32 elements and set B has 42 elements and A       B contains 62 elements, find the
number of elements in A B.




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                                                     Set Theory


                                 REVIEW – EXTRA PRACTICE
Group 28: Answer each group of questions below.

1. If D = {0, 4, 7}, we say that 7  D, or {7}  D, but cannot say 7  D, since 7 is not a set.
Which of the following are true?

         a.   4D                                                 f. 0  D
         b.   4 D                                                g. 4 = {4}
         c.   0D                                                 h. 4  {4}
         d.    D                                                i. 0 = 
         e.     D                                               j. 0  

2. If A = {a, b, c} and B = {a, b, c, d}, which of the following are true?

         a.   AB                                                 f.   B
         b.   A B                                                g.   A
         c.   aA                                                 h. a  B
         d.   bB                                                 i. B  A
         e.   b B

3. If A  X, b  Y, X  Z, and Y  Z, is:

         a. a  Z
         b. b  Z
         c. a  Y
         d. Can there be an element in Z which is an element of both X and Y?
         e. Can there be an element in Z which is an element of X but not Y?
         f. Can there be an element in Z which is neither an element of X nor of Y?


Group 29: Answer each group of questions below.

1. Tell under what conditions on the sets A and B we would have each of the following.

a. A     B= 
b. A     B=U
c. A     B=U
d. A     B= 
e. A     B=A
f. A     B=A
g. A     = 
h. A     U=A
i. A     U=U



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                                                     Set Theory

j. A U = A
k. A  = U
l. A  = 


2. Tell whether each of the following statements is true or false for any two sets A and B, if
B  A.

         a.   A    B is always equal to A.
         b.   A    B is always equal to B.
         c.   A    B is always equal to A.
         d.   A    B is always equal to B.


Group 30: Fill in the diagram below using all the members of these sets.

         A = {1, 2, 3, 4, 5, 6, 7}
         B = {1, 3, 5, 7, 9}
         C = {3, 4, 5, 6, 7}
         D = 0, 1, 3, 4}




D:\Docstoc\Working\pdf\2d746796-c1a8-49f3-ae9b-7b28ea72e29c.doc

								
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