2023-11-15

CS1029/notes-2023-11-01

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+Edge contractions: merge two vertices, removing an edge between them
+Representations: adjacency list
+				 adjacency matrix
+					What does the determinant of an adjacency matrix mean?
+				incidence matrix: vertices against edges, 1 where edge is connected to vertex
+

CS1029/notes-2023-11-07

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+

CS1029/notes-2023-11-08

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+

CS1029/notes-2023-11-14

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+

CS1029/practical-2023-11-03/relations

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

CS1029/practical-2023-11-10/graphs

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+1. Undirected, unlooped, multi-edged: multigraph
+b) directed, looped, multi-edged: directed pseudo-multigraph
+
+2. ac bd
+b) cd cd dd ee ab bc
+
+3. {{paper}}
+
+4. vertices: 6
+   edges: 6
+   degree: a: 2 b: 4 c: 1 f: 3 e: 2 d: 0
+   isolated: d
+   pendant: c
+b) vertices: 5
+   edges: 14
+   degree: a: 6 b: 6 c: 6 d: 5 e: 3
+   isolated: -
+   pendant: -
+
+5. vertices: 4
+   in-a : 2
+   out-a: 2
+   in-b:  3
+   out-b: 4
+   in-c:  2
+   out-c: 1
+   in-d:  1
+   out-d: 1
+
+6. {ac} {bde}
+b) Not bipartite: 3-loop bcf would require 3 sets
+
+7. {{ paper }}
+
+8. a -> abcd
+   b ->    d
+   c -> ab  
+   d ->  bcd
+
+9.    | a b c d
+    --+--------
+    a | 1 1 1 1
+    b | 0 0 0 1
+    c | 1 1 0 0
+    d | 0 1 1 1
+
+10. {{ paper }}
+
+11. v1 -> u1
+    v2 -> u4
+	v3 -> u2
+	v4 -> u5
+	v5 -> u3
+
+12. v1 -> u4
+	v2 -> u3
+	v3 -> u1
+	v4 -> u2
+
+13. PSCL
+ a) YNN4
+ b) N---
+ c) N---
+ d) YYY5