feat: w1 st2334 notes

.gitmodules

@@ -4,3 +4,6 @@
 [submodule "cs4212/hw1-hellocaml-yadunut"]
 	path = cs4212/hw1-hellocaml-yadunut
 	url = https://github.com/cs4212/hw1-hellocaml-yadunut
+[submodule "cs4212/week-02-x86lite"]
+	path = cs4212/week-02-x86lite
+	url = https://github.com/cs4212/week-02-x86lite

cs3231/main.typ

@@ -30,3 +30,15 @@
   - $L_1 dot L_2 = L_1L_2 = {x y : x in L_1, y in L_2}$
   - $L^* = {x_1x_2...x_n : x_1, x_2, ... x_n in L, n in NN}$
   - $L^+ = {x_1x_2...x_n : x_1, x_2, ... x_n in L, n gt.eq 1}$
+
+=== Proof Structure
+==== Set A = Set B
+1. Show $A subset.eq B$
+  - Take *arbitrary* element $x in A$
+  - Use definition of A to show $x in B$
+  - Therefore, $A subset.eq B$
+2. Show $B subset.eq A$
+  - Take *arbitrary* element $x in B$
+  - Use definition of B to show $x in A$
+  - Therefore, $B subset.eq A$
+3. Since $A subset.eq B$ and $B subset.eq A$, $A = B$

cs4212/hw1-hellocaml-yadunut

@@ -1 +1 @@
-Subproject commit a976b25ebb1434189ba15627bab1720fb345f518
+Subproject commit a152b7a34c9b2038b1588a3e28a5d156d41db0bf

cs4212/week-02-x86lite

@@ -0,0 +1 @@
+Subproject commit 4d81e478221715bf176adefc647f8b668d4ff4dc

st2334/main.typ

@@ -0,0 +1,65 @@
+#import "@preview/ilm:1.4.1": *
+
+
+// #show: ilm.with(
+//   title: [],
+//   author: "Yadunand Prem",
+//   table-of-contents: none,
+// )
+#set text(lang: "en", font: ("SF Pro Display"))
+#set heading(numbering: "1.")
+#show raw: set text(font: "SF Mono")
+
+= Reference
+- Sample Space ($S$): Set of all possible outcomes of an experiment
+  - Can vary depending on the problem of interest
+- Sample Point: Outcome of sample Space (Element)
+- Event: Subset of sample space (Set)
+
+== Set Operations
+- $A union B = {x : x in A "or" x in B}$
+- $A inter B = {x : x in A "and" x in B}$
+- $A' = {x : x in S "and" x in.not A}$
+- $limits(union.big)^n_(i=1) A_i = A_1 union A_2 union ... union A_n = {x: x in A_1 "or" " ... or"  x in A_n}$
+- Mutually exclusive / disjoint - $A inter B = emptyset$
+- Contained - All in A are also elements in B, A is contained in B, $A subset B$ or $B supset A$
+- Equivalent - $A subset B$ and $B subset A$, then $A = B$
+
+== Set Operations
+- $A inter A' = emptyset$
+- $A inter emptyset = emptyset$
+- $A union A' = S$
+- $(A')' = A$
+- $A union (B inter C) = (A union B) inter (A union C)$
+- $A inter (B union C) = (A inter B) union (A inter C)$
+- $A union B = A union (B inter A')$
+- $A = (A inter B)  union (A inter B')$
+- $(A_1 union A_2 union ... union A_n)' = A_1^' inter A_2^' inter ... inter A_n^'$
+- $(A_1 inter A_2 inter ... inter A_n)' = A_1^' union A_2^' union ... union A_n^'$
+
+== Counting Methods
+- Multiplication Principle - $r$ different experiments to be performed sequentially. Then there are $n_1n_2...n_r$ possible outcomes for $r$ experiments
+- Addition Principle - experiment can be performed by $k$ different procedures. Suppose ways under different procedures *do not overlap*. Then total is $n_1 + ... + n_k$
+- *Permutation* is selection of $r$ objects out of $n$. Order is taken into consideration.
+  $ P^n_r = n! / (n-r)! = n(n-1)(n-2)...(n-(r-1)) $ (When $r = n, P^n_n = n!$)
+- *Combination* is selection of $r$ objects out of $n$, without regard for order.
+  $ vec(n, r) = n!/(r!(n-r!)) = vec(n, n-r) $
+  - Intuition: In terms of permutation, no of ways to choose and arrange $r$ objects out fo $n$ is $P_r^n$
+  - This can be also done by the following:
+    - Select $r$ objects out of $n$ without regard to order: $vec(n, r)$ ways
+    - For each combination, permute its $r$ objects: $P^r_r$ ways
+    - For each combination, permute its $r$ objects: $P^r_r$ ways
+    - $vec(n, r) times P^r_r = P^n_r$
+
+== Probability
+- Probability is chance or how likely a certain event may occur. Let $A$ be an event in an experiment. $P(A)$ is to quantify how likely $A$ may occur.
+=== Axioms
+Probability, $P(dot)$ is a function on the collection of events in the sample space satisfying:
+- For any event $A$, $0 lt.eq P(A) lt.eq 1$
+- For the sample space $P(S) = 1$
+- For any 2 mutually exclusive event $A$ and $B$, that is $A inter B = emptyset$, $P(A union B) = P(A) + P(B)$
+- $P(emptyset) = 0$
+- $P(A') = 1-P(A)$
+- $P(A) = P(A inter B) + P(A inter B')$
+- $P(A union B) = P(A) + P(B) - P(A inter B)$
+- $A subset B$, then $P(A) lt.eq P(B)$