59 lines
1.8 KiB
Plaintext
59 lines
1.8 KiB
Plaintext
1. a) {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 4), (4, 5), (5, 5) }
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Reflexive: \forall x. (x, x) \in R
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Antisymmetric (& not symmetric): all pairs have x <= y. forall x, y. x <= y, !(y <= x) || y == x
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Transitive
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b) {(1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 4), (4, 1), (4, 3)}
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Not reflexive: there are no pairs of (x, x)
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Symmetric - all pairs have their reverse represented
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Not antisymmetric: symmetric and anti-symmetric are mutually exclusive
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Not transitive: (1, 2) and (2, 3) - 1 + 3 is not odd
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2. {(item, quantity)}
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{(Name, {(key, value)})}
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{(Name, Address, {(Room type, price, {(key, value)})}, ...)}
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3. 105 305 306 505 705 707 905 906 909
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4. a) ab ac bc cb
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b) {(a, a), (a, b), (a, c), (b, b), (b, a), (b, c), (c, a), (c, b), (d, d)}
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5.a) {(1, 1), (1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4)}
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Not reflexive: (2, 2) is not present
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Symmetric, therefore not anti-symmetric
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Not transitive: (1, 2) and (2, 3), but not (1, 3)
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b) 12 21 14 41 32 23 43 34
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Not reflexive
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Symmetric, therefore not antisymmetric
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Not transitive: 12 and 23 but not 13
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6. 1 1 0 0
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1 1 0 0
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1 0 1 1
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0 0 0 1
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Yes, it's reflexive
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8. {(a, b) | a divides b OR b divides a}
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9. No, it's not transitive. (a, b) & (b, d), but not (a, d)
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10. a) Not equivalence relation: missing transitivity
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(1, 3) and (3, 2), but not (1, 2)
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b) {0}, {1, 2}, {3}
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11. \forall n \in N_0:
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0 + 3n
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1 + 3n
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2 + 3n
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12. a) Y
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b) N: 0 is in both - not disjoint
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c) Y
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d) N: 0 is missing
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13. a) 00 11 22 33 44 55 12 21 34 43 35 53 45 54
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b) 00 11 22 33 44 55 01 10 23 32 45 54
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c) 00 11 22 33 44 55 01 10 02 20 12 21 34 43 35 53 45 54
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14. a) Y, trivially
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b) N: not antisymmetric ((2, 3) and (3, 2))
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c) N: not reflexive (no (3, 3))
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