notes

commit ccc280ed7d70e37089abf4d531e5150687fa6aab
parent 30b8ee6f676ed1b3657abd1c48114886fcf0d9d9
Author: Andrew <andrewlaack1@gmail.com>
Date:   Wed, 21 Aug 2024 06:11:31 -0500

discrete notes

Diffstat:
A
Cases.md
 | 
8
++++++++
M
DiscreteMath.md
 | 
6
++++++
A
ExhaustiveProof.md
 | 
10
++++++++++
A
Set.md
 | 
32
++++++++++++++++++++++++++++++++
A
VacuousProof.md
 | 
8
++++++++
M
index.md
 | 
4
----

6 files changed, 64 insertions(+), 4 deletions(-)
diff --git a/Cases.md b/Cases.md
@@ -0,0 +1,8 @@
+:discrete: :logic: :proof: 
+# Proof by Cases
+
+U 1.8.1
+
+## Notes
+
+**Definition:** Proof by cases is a form of proof whereby we show every specific type of case is true.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -49,3 +49,9 @@ Unit 1.7:
 	- [[DirectProof.md]]
 	- [[Contrapositive.md]] - Also known as contraposition
 	- [[Contradiction.md]]
+	- [[Cases.md]]
+	- [[VacuousProof.md]]
+	- [[ExhaustiveProof.md]]
+
+Unit 2.1:
+	- [[Set.md]]
diff --git a/ExhaustiveProof.md b/ExhaustiveProof.md
@@ -0,0 +1,10 @@
+:discrete: :proof: :logic:
+# Exhaustive Proof
+
+U 1.8.2
+
+## Notes
+
+**Definition:** An exhaustive proof is similar to proof by cases except we evaluate it for all specific examples which needs to be a relatively small number.
+
+An exhaustive proof that "a < a+1 for 1 < a < 5 where a $\in \Z$" would show the statement is true for 2, 3, and 4.
diff --git a/Set.md b/Set.md
@@ -0,0 +1,32 @@
+:discrete: :math:
+# Set
+
+U 2.1.1
+
+## Notes
+
+**Definition:** A set is an unordered list of elements.
+
+Common Sets:
+
+$\N$ - Natural Numbers
+
+$\mathbb{W}$ - Whole Numbers (0 + Natural Numbers)
+
+$\Z$ - Integers
+
+$\mathbb{Q}$ - Rational Numbers ($\frac{a}{b}, b \neq 0$) 
+
+$\R$ - Real numbers (not imaginary)
+
+$\mathbb{C}$ - Complex numbers (of the form a + bi)
+
+## Specifications
+
+Set specification types:
+
+1. Roster - list of all members (discrete)
+2. Set Builder - specify all members by form (discrete or continuous)
+3. Interval Notation - specify set as [a,b) where a is inclusive and b is exclusive (continuous)
+
+Inverval notation is best for continuous sets where it is applicable, otherwise we generally use set builder notation unless there is a really small cardinality of the specified set.
diff --git a/VacuousProof.md b/VacuousProof.md
@@ -0,0 +1,8 @@
+:discrete: :logic: :proof: 
+# Vacuous Proof
+
+U 1.7
+
+## Notes
+
+**Definition:** A vacuous proof is for proofs of the form if p then q where we then show that p is always false thus there is no need to evaluate for q.
diff --git a/index.md b/index.md
@@ -36,10 +36,6 @@ Stats + Prob
 	- [ ] https://en.wikipedia.org/wiki/Continuous-time_Markov_chain
 
 Lin-alg
-	- [ ] Implement Stochastic Approach to Eigen vector/value calculation using grid.
-	- [ ] Inverse Matrix Calculations (do some practice problems)
-	- [ ] https://en.wikipedia.org/wiki/Invertible_matrix
-	- [ ] Laplace Expansion (define outside of determinant note)
 	- [ ] Cofactor (define)
 	- [ ] Minors