ch3ans.pdf - 3.3 Exercises 1 For the people in the family tree(see Figure 3-2 build tables for the following relations(a IsAncestorOf(b IsDescendentOf(c

ch3ans.pdf - 3.3 Exercises 1 For the people in the family...

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3.3 Exercises 1. For the people in the family tree (see Figure 3-2), build tables for the following relations: (a) IsAncestorOf (b) IsDescendentOf (c) IsSiblingOf (d) IsCousinOf 1. = John Mary Maude=Harold Peter=Elaine George Elizabeth 1. (a) IsAncestorOf Elaine George Mary George John George Peter George Harold Elizabeth Maude Elizabeth Mary Elizabeth John Elizabeth Mary Elaine John Elaine Mary Maude John Maude 1. (b)
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IsDescendentOf George Mary George John George Peter George Elaine Maude Mary Maude John Elizabeth Maude Elizabeth Harold Elizabeth Mary Elizabeth John Elaine Mary Elaine John 1. (c) IsSiblingOf Elaine Maude Maude Elaine 1. (d) IsCousinOf George Elizabeth Elizabeth George 3. Find the elements in each of the following relations defined on R : (a) ( x, y ) R if and only if x + 1 < y (b) ( x, y ) R if and only if y < 0 or 2 x 3 (c) ( x, y, z ) R if and only if x 2 + y = z 3. (a) 3. (b)
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3. (c) -2 0 2 -2 0 2 0 5 10 5. The table gives the names of airlines and several cities that each flies to from Chicago. The table also gives the number of miles for each flight. List all the triples (X, Y, Z) of the ternary relation defined by those triples for which airline X flies Y miles to city Z . TWA Pan Am Piedmont Topeka 603 Bombay 7809 Peoria 170 Kansas City 510 Seattle 2052 Albany 816 Phoenix 1742 Anaheim 2025 Atlanta 717 5. Airlines Mileage City TWA 603 Topeka TWA 510 Kansas City TWA 1742 Phoenix Pan Am 7809 Bombay Pan Am 2052 Seattle Pan Am 2025 Anaheim Piedmont 170 Peoria Piedmont 816 Albany Piedmont 717 Atlanta
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7. Using the family tree shown in Figure 3-2, list the elements in each of the following relations, and give these relations meaningful names. (a) IsMarriedTo - 1 (b) IsMarriedTo IsMarriedTo (c) IsParentOf IsParentOf - 1 (d) = Family where Family denotes the set of people appearing in the family tree (e) IsMarriedTo IsMarriedTo - 1 (f) IsParentOf IsParentOf - 1 7. (a) IsMarriedTo - 1 John Mary Mary John Peter Elaine Elaine Peter Maude Harold Harold Maude 7. (b) IsMarriedTo IsMarriedTo Peter Peter Elaine Elaine Maude Maude Harold Harold Mary Mary John John 7. (c) IsParentOf IsParentOf - 1 Elaine Elaine Elaine Maude Maude Elaine Maude Maude George George Elizabeth Elizabeth
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7. (d) = Family Peter Peter Elaine Elaine Mary Mary John John Maude Maude Harold Harold Elizabeth Elizabeth George George 7. (e) IsMarriedTo IsMarriedTo - 1 John Mary Mary John Peter Elaine Elaine Peter Maude Harold Harold Maude 7. (f) 9. Prove Theorem 2(c). 9. For any x, y X such that ( x, y ) S - 1 , it follows that ( y, x ) S. Since S R, ( y, x ) R. Now, ( x, y ) R - 1 . 11. Let A = { 1 , 2 , 3 , . . ., 10 } . Let R = { (1, 2), (1, 4), (1, 6), (1, 8), (1, 10), (3, 5), (3, 7), (4, 6), (6, 8), (7, 10) } be a relation on A. Let S = { (2 , 4) , (3 , 6) , (5 , 7) , (7 , 9) , (8 , 10) , (8 , 9) , (8 , 8) , (9 , 9) , (3 , 8) , (4 , 9) } be a sec- ond relation on A. Find: (a) R S (b) S R. 11. (a) { (2 , 6) , (3 , 8) , (5 , 10) } 11. (b) { (1 , 4) , (1 , 9) , (1 , 10) , (1 , 8) , (3 , 7) , (3 , 9) , (6 , 10) , (6 , 9) , (6 , 8) }
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13. Let R, S and T be binary relations on a set X .
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