Initial work

CS1029/notes-2023-09-19

@@ -0,0 +1,6 @@
+Test in late October & late November, final exams in December (20%, 20%, 80%); all multiple choice?
+Books: (reccommended)
+  - Discrete Mathematics and Its Applications, 8th edition,Kenneth H. Rosen
+  - Mathematics for Calculus, 6th edition. Stewart et al.
+
+

CS1029/notes-2023-09-20

@@ -0,0 +1,4 @@
+Why must we review *rounding*, how negatives multiply, assoc/commut/distrib properties - all of this is early secondary school or primary school!
+  exponentiation, absolute values
+
+  Is this not assumed knowledge?!

CS1029/notes-2023-09-26

@@ -0,0 +1,2 @@
+# Propositional Logic
+

CS1029/notes-2023-10-11

@@ -0,0 +1,8 @@
+image = range
+codomain: set that a function is stated to evaluate to; range: values that are possible from a function
+onto function: surjection
+bijective: injective & surjective
+injective: \forall x, y \in \operatorname{Domain}.~ f(x) = f(y) \implies x = y \tag{one-to-one}
+surjective: \forall y \in \operatorname{Codomain}.~\exists x \in \operatorname{Domain}.~ f(x) = y \tag{codomain = range}
+
+inverse only a function for bijective functions - not surjective means undefined for some values of the domain, not injective means multi-valued for some values of the domain 

CS1029/notes-2023-10-17

@@ -0,0 +1 @@
+

CS1029/notes-2023-10-18

@@ -0,0 +1 @@
+

CS1029/notes-2023-10-25

@@ -0,0 +1,7 @@
+Closure of a relation R is the smallest relation containing R and meeting a property
+	EG. reflexive closure of R (on A) is R | { (a, a) | a \in A }
+	symmetric closure of R is R | mirror(R)
+
+Equivalent: R is reflexive, symmetric, and transitive - f(x) = f(y) for some function f
+Equivalence classes: sets of values which R considers equivalent
+	disjoint, nonempty subsets of domain

CS1029/notes-2023-10-31

@@ -0,0 +1,13 @@
+Simple Graph: unique connections, no reflexive connections
+Multigraphs: multiple connections between two vertices permitted
+Directed graphs/digraph
+Pseudograph: permits self-links
+Degree of vertex in an undirected graph: edges connected to vertex (loops contribute twice)
+                 in a directed graph: in-degree is edges pointing to a vertex, out-degree is pointing out
+
+Pendant vertex has degree 1
+Sum of the degree of all vertices in an undirected graphs is twice the number of edges: handshake theorem
+Bipartite: can be partitioned into two sets of vertices such that no edge connects two vertices in the same set
+	Complete bipartite: two vertices are connected \iff they are in seperate partitions
+
+	Matching in a bipartite graph: find a subgraph such that all vertices have *exactly* 1 edge attached

CS1029/practical-2023-09-29/datasci

@@ -0,0 +1,11 @@
+Books: id.loc.gov
+  Library of Congress linked data
+  Already in relatively usable formats: XML, JSONLD, etc; further work would be dependent on type of analysis
+
+Disease risk: data.gov.ie - large amounts of discharge data in Health/HSE section
+  Something could be stitched together estimating risks
+
+Universities: https://www.ucas.com/data-and-analysis/undergraduate-statistics-and-reports
+  All CSVs in zips
+
+

CS1029/practical-2023-09-29/numbers

@@ -0,0 +1,6 @@
+06. 1 0 1/2 π
+07. commutativity (+), commutativity (*), associativity (+), associativity (*), distributivity of * over +
+08. 3x+3y 8a-8b 28y-14x 3ab+3ac-6ad
+09. 17/30 9/20 3 1/36
+11. 100 -12 6/24 2 -1 1
+13. 109.9884 48.36 30.24 42313990.36

CS1029/practical-2023-10-06/prop

@@ -0,0 +1,93 @@
+01. TFTF----FF
+02. a) Linda is older than Sanjay (given that age is continuous)
+   b) Mei does not make more money than Isabella (salaries are discrete)
+   c) Moshe is shorter than Monica (height is continuous)
+   d) Abby is not richer than Ricardo (is wealth continuous?)
+   e) Quincy is not smarter than Venkat (unquantifiable ∴ continuous)
+   f) 2 + 1 /= 3
+   g) The summer in Maine is not hot and sunny
+   i) Jennifer and Teja are not friends
+03. TTFF
+4 a) I did not buy a lottery ticket this week
+  b) I bought a lottery ticket this week and I won the million dollar jackpot
+  c) If I bought a lottery ticket this week, I won the million dollar jackpot
+  d) I bought a lottery ticket this week, or I won the million dollar jackpot
+  e) I bought a lottery ticket if and only if I won the million dollar jackpot
+  f) If I didn't buy a lottery ticket this week, I didn't win the million dollar jackpot
+  g) I didn't buy a lottery ticket this week, and I didn't win the million dollar jackpot
+  h) I didn't buy a lottery ticket this week, or I did buy a lottery ticket this week and I won the million dollar jackpot
+  -> If I bought a lottery ticket this week, I won the million dollar jackpot
+05. a) !p
+   b) p & !q
+   c) p -> q
+   d) !p -> !q
+   e) p -> q
+   f) q & !p
+   g) p <-> q
+06. a) r & !p
+   b) !p & q & r
+   c) r -> (!p <-> q)
+   d) !q & !p & r
+   e) q -> (!r & !p)
+   f) (p & r) -> !q
+07. TFTF
+08. FTTT
+09. a) If one does not wash the boss's car, one is not promoted
+   b) If there exist winds from the south, there will exist a spring thaw
+   c) If the computer was bought less than a year ago, the warranty is good
+   d) If Willy cheats, he is caught
+   e) If you do not pay a subscription fee, you cannot access the website
+   f) If one knows the right people, one is elected
+   g) If Carol is on a boat, she gets seasick
+10. a) p  !p   p -> !p
+       T   F     F
+       F   T     T
+    b) p  !p   p <-> !p
+       T   F     F
+       F   T     F
+    c) p   q   p ^ (p | q) (== !p & q)
+       T   T     F
+       T   F     F
+       F   T     T
+       F   F     F
+    d) p   q   (p | q) -> (p & q)
+       T   T     T
+       T   F     F
+       F   T     T
+       F   F     T
+11. OR           AND          XOR
+    111_1111     000_0000     111_1111
+    1111_1010    1010_0000    0101_1010
+    10_0111_1001 00_0100_0000 10_0011_1001
+12. a) 11000
+    b) 10001
+
+13. 
+ a) p q r  (p | q) ((p | q) | r) (q | r) (p | (q | r))
+    T T T     T          T          T         T
+    T T F     T          T          T         T
+    T F T     T          T          T         T
+    T F F     T          T          F         T
+    F T T     T          T          T         T
+    F T F     T          T          T         T
+    F F T     F          T          T         T
+    F F F     F          F          F         F
+ b) p q r  (p & q) ((p & q) & r) (q & r) (p & (q & r))
+    T T T     T          T          T         T
+    T T F     T          F          F         F
+    T F T     F          F          F         F
+    T F F     F          F          F         F
+    F T T     F          F          T         F
+    F T F     F          F          F         F
+    F F T     F          F          F         F
+    F F F     F          F          F         F
+
+14. a) Jan is not rich or is not happy
+    b) Carlos will not run and will not bicycle tomorrow
+    c) Mei does not walk and does not take the bus to class
+    d) Ibrahim is not smart or is not hard-working
+    
+15. (p -> q) & (p -> r)
+ ≣ (!p | q) & (!p | r) # Definition of ->
+ ≣ (!p | (q & r)) # Distribution of | over &
+ ≣ p -> (q & r) # Definition of ->

CS1029/practical-2023-10-13/sets

@@ -0,0 +1,46 @@
+1. a) the students who either live within 1 mile of the school or who walk to classes
+   b) the students who live within 1 mile of the school and walk to classes
+   c) the students who live within 1 mile of the school and do not walk to classes
+   d) the students who walk to classes and do not live within 1 mile of the school
+
+2. a) { 0, 1, 2, 3, 4, 5, 6 }
+   b) { 3 }
+   c) { 1, 2, 4, 5 }
+   d) { 0, 6 }
+
+3. a) { a, b, c, d, e, f, g, h }
+   b) { a, b, c, d, e }
+   c) {}
+   d) { f, g, h }
+
+5. A \union B = { x | x in A || x in B }; commutative as OR is commutative
+   A \intersect B = { a | x in A && x in B }; commutative as AND is commutative
+
+6. A - B = { x | x \in A \and x \nin B }
+   !B = { x | x \nin B }
+   A & !B = { x | x \in A \and x \in !B } = { x | x \in A \and x \nin B }
+
+   (A & B) | (A & !B) = (A | A) & (B | !B) (distribution of & over |)
+   = A & U
+   = A
+
+7. a) { 4 6 }
+   b) { 0 1 2 3 4 5 6 7 8 9 10 }
+   c) { 4 5 6 8 10 }
+   d) { 0 2 4 5 6 7 8 9 10 }
+
+9. { 2 5 }
+
+11. A ^ B = { x | x in A ^ x in B } = { x | (x in A | x in B) & !(x in A & x in B) }
+          = { x | x in A | x in B } - { x | x in A & x in B }
+		  = (A | B) - (A & B)
+
+12. a) { a: 3, b: 3, c: 1, d: 4 }
+    b) { a: 2, b: 2 }
+	c) { a: 1, c: 1 }
+	d) { b: 1, d: 4 }
+	e) { a: 5, b: 5, c: 1, d: 4}
+
+14. A_i = { x | x in Z, x <= i }
+    a) A_1
+	b) A_n

CS1029/practical-2023-10-20/functions

@@ -0,0 +1,52 @@
+1. a) f(x) is undefined for x = 0
+   b) f(x) is complex for x < 0
+   c) multivalued
+
+2. a) multivalued
+   b) It is
+   c) undefined at x = 2
+
+3. a) Z >= 0, {0..9}
+   b) 0.. -> 1..?
+   c) All bit strings -> N
+   d) All bit strings -> N
+   e) (Z^+)^2 -> Z^+
+   f) (Z^+)^2 -> Z^+
+
+4. a) 1 b) 2 c) -1 d) 0 e) 2 f) 3 g) 0 h) 2
+5. a) 1 b) 0 c) 0 d) -1 e) 3 f) -1 g) 2 h) 1
+
+8. a) 6 b) 24 c) 120 d) 3_628_800
+
+10. a)y b)n c)n
+11. {a}
+
+12. a) y b) n c) n d) n
+13. a) y b) n) c) y) d) n
+
+14. ynyy
+
+15. a) everyone has a phone number (phone numbers should already be uniquie)
+    b) None (student ids should already be unique)
+	c) Every student gets a unique grade
+	d) no two people come from the same town
+
+16. a) the set of phone numbers
+    b) the set of student ids
+	c) 1..100
+	d) all towns
+
+17. ynnn
+18. a) { 1 }
+    b) { -1, 1, 5, 9, 15 }
+	c) { 0, 1, 2 }
+	d) { 0, 1, 2 }
+
+19. f.g = x^2 + 4x + 5, g.f = x^2 + 3
+20. f + g = x^2 + x + 3
+    fg = x^3 + 2x^2 + x + 2
+
+21. f^-1 doesn't exist, assuming f^-1(x) = +sqrt(x)
+   a) { 1 }
+   b) { x | 0 < x < 1 }
+   c) { x | x > 2 }

CS1029/practical-2023-10-20/m.py

@@ -0,0 +1,22 @@
+def floor(n):
+	return n - (n % 1)
+
+def ceil(n):
+	return n - (n % -1)
+
+
+assert floor(-1) == -1
+assert floor(1) == 1
+assert floor(1.2) == 1
+assert floor(-1.2) == -2
+assert floor(0) == 0
+assert floor(0.1) == 0
+assert floor(-0.1) == -1
+assert ceil(-1) == -1
+assert ceil(1) == 1
+assert ceil(1.2) == 2
+assert ceil(-1.2) == -1
+assert ceil(0) == 0
+assert ceil(0.1) == 1
+assert ceil(-0.1) == 0
+

CS1029/practical-2023-10-20/sequences

@@ -0,0 +1,35 @@
+1. a) 3
+   b) -1
+   c) 787
+   d) 2639
+
+2. a) 128
+   b) 7
+   c) 2
+   d) -256
+
+3. a) 1, -2, 4, -8
+   b) 3, 3, 3, 3
+   c) 8, 13, 23, 71
+   d) 2, 0, 8, 0
+
+4. a) 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
+   b) 0, 0, 0, 1, 1, 1, 2, 2, 2, 3
+   c) 1, 1, 3, 3, 5, 5, 7, 7, 9, 9
+   d) 3, 6, 12, 24, 48, 96, 192, 384, 768, 1336
+
+5. a_0 = 3, a_n = a_{n - 1} + 2 (i.e. the odd integers starting at 3)
+   the primes starting at three
+   - one more
+
+6. a) 2, 12, 72, 432, 2592
+   b) 2, 4, 16, 256, 65536
+   c) 1, 2, 5, 11, 26
+   d) 1, 1, 6, 27, 204
+
+7. a) a_n = -3a_{n-1} + 4a_{n-2} = -3(0) + 4(0) = 0 = a_n
+   b) a_n = -3(1) + 4(1) = 4 - 3 = 1 = a_n
+   c) a_n = -3((-4)^{n-1}) + 4((-4)^{n-2}) = -3(-4)(-4^{n-2}) + 4((-4)^{n-2}) = (-4)^{n-2}((-3)(-4) + 4) = (-4)^{n-1}(-3 - 1) = (-4)^n = a_n
+   d) a_n = -3(2*(-4)^{n-1} + 3) + 4(2 * (-4)^{n-2} + 3) = -6(-4)^{n-1} - 9 - 2(-4)^{n-1} + 12 = -8(-4)^{n-1} + 3 = 2(-4)^n + 3 = a_n
+
+8.