Solve these problems:
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= EXERCISES 5.1 1. For each of the following binary relations p on N, decide which of the given ordered pairs belong to p. a. xpy + x + y<7; (1, 3), (2,5), (3, 3), (4,4) b. xpy + x = y + 2; (0,2), (4,2), (6, 3), (5,3) c. xpy + 2x + 3y = 10; (5,0), (2, 2), (3, 1), (1,3) d. xpy →y is a perfect square; (1, 1), (4,2), (3,9), (25,5) 2. For each of the following binary relations p on Z, decide which of the given ordered pairs belong to p. a. xpy + x[y; (2, -6), (3,5), (8,4), (4,8) b. xpy x and y are are relatively prime; (5,8), (9, 16), (6, 8), (8, 21) c. xpy → ged(x, y) = 7; (28, 14), (7, 7), (10,5), (21, 14) d. xpy or + y2 = 2+ for some integer z; (1, 0), (3,9), (2, 2), (-3, 4) e. xpy x is a number from the Fibonacci sequence; (4, 3), (7,6), (7, 12), (20, 20) 3. Decide which of the given items satisfy the relation. a. p a binary relation on Z, xpy - x = -y; (1, - 1), (2, 2), (-3, 3), (-4,-4) b. p a binary relation on N, xpy + x is prime; (19, 7), (21, 4), (33, 13), (41, 16) c. p a binary relation on Q, xpy x S 1/y; (1, 2), (-3, -5),(-4,1/2), (1/2, 1/3) d. p a binary relation on NXN, (x, y)p (u, v) +x+u = y + v; (1, 2), (3, 2)), ((4, 5), (0, 1))
4. Decide which of the given items satisfy the relation. a. pabinary relation on S X C, where S = {states in the United States}, C = {cities in the United States), xpyhy is the capital of xj (Indiana, Indianapolis), (Illinois, Chicago), (Kansas, Kansas City), (Kentucky, Louisville), (North Dakota, Bismarck) b. p a binary relation on A X P, where A = {artists}, P = {paintings}, xpy + x painted y; (DaVinci, Mona Lisa), (Grant Wood, American Gothic), (Remington, Ridden Down), (Picasso, Blue Dancers), (van Gogh, Starry Night) c. p a binary relation on C X M, where C = {composers), M = {music}, xpy -x composed y; (Bernstein, West Side Story), (Presley, Blue Suede Shoes), (Gershwin, Rhapsody in Blue), (Beethoven, Moonlight Sonata), (Rogers and Hammerstein, Phantom of the Opera) d. p a binary relation on A X B, where A = {authors}, B = {books}, xpy or wrote y; (Hemingway, The Old Man and the Sea), (Sawyer, Huckleberry Finn) (Poe, Moby Dick), (Orwell, 1984), (Tolstoy, Crime and Punishment)