commit c5894452dceb794bf3161e3738e43995bd5b7f2f
parent 57eea0312c3188eda3eb76c19cdbe24c02941239
Author: Andrew <andrewlaack1@gmail.com>
Date: Sun, 18 Aug 2024 23:47:54 -0500
Took notes on discrete math
Diffstat:
9 files changed, 94 insertions(+), 3 deletions(-)
diff --git a/DemorgansLaw.md b/DemorgansLaw.md
@@ -38,3 +38,7 @@ The first law states that not p and q is the same as not p or not q.
The second law states that not p or q is the same is not p and not q.
This is basically the distributive property of boolean logic whereby we flip the and/or connective and distribute the negation.
+
+#### For Quantifiers
+
+See [[Quantifiers.md]] section on negation which describes the distribution of a negation when quantifiers are involved.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -30,3 +30,8 @@ Unit 1.3:
- [[WellDefined.md]]
- [[Commutative.md]]
- [[Satisfiable.md]]
+
+Unit 1.4:
+ - [[Predicate.md]]
+ - [[Quantifiers.md]]
+ - [[Universe.md]]
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -72,4 +72,6 @@ Khan Unit 2:
- [[GaussianElimination.md]]
- [[EigenVector.md]]
- [[Transpose.md]]
+
+Khan Unit 3:
- [[OrthogonalComplement.md]]
diff --git a/OrthogonalComplement.md b/OrthogonalComplement.md
@@ -12,3 +12,11 @@ The orthogonal complement of the subspace V in $\R^n$ is defined as follows:
$V^\perp = \{\vec{x} \in \R^n | \vec{x} \cdot \vec{v} = 0 \text{ and } \vec{v} \in V \}$
The orthogonal complement of a subspace is a subspace in all cases as it respects scalar multiplication, vector addition, and contains the zero vector.
+
+Every element of the nullspace is in the orthogonal complement and vice versa thus they are the same set.
+
+## Dimensionality
+
+For the arbitrary subspace V, we know dim(V) = k. As such, we also know for O which is the orthogonal complement, that dim(O) = k - n where R^n is the [[AmbientSpace.md]].
+
+This is given because we also know that the [Nullity](Nullity.md) + [Rank](Rank.md) = dim([AmbientSpace](AmbientSpace.md)).
diff --git a/Predicate.md b/Predicate.md
@@ -0,0 +1,24 @@
+:discrete: :math: :logic:
+# Predicate
+
+U 1.4.1
+
+## Notes
+
+**Definition:** The predicate in a mathematical context is the part of a statement that gives us a truth value.
+
+In the case of 'x < 2' the predicate is 'less than 2'. This can be stated as a propositional function P(x). The following are valid inputs and outputs of said function:
+
+P(1) = True
+
+P(2) = False
+
+P(3) = False
+
+Another example is 'x + y = z' denoted as R(x,y,z) where the predicate is '='.
+
+R(2, -1, 5) = False
+
+R(3, 4, 7) = True
+
+R(x, 3, z) not a proposition, but R is still a propositional function.
diff --git a/Quantifiers.md b/Quantifiers.md
@@ -0,0 +1,37 @@
+:logic: :math: :discrete:
+# Quantifiers
+
+U 1.4.2
+
+## Notes
+
+**Definition:** Quantifiers are operators that describe the number of individuals in a domain that satisfy something.
+
+The two common quantifiers are:
+
+1. $\exists$ - There exists (existential quantifier)
+2. $\forall$ - For all (universal quantifier)
+
+Another derived one is $\exists !$ which means there exists only one.
+
+### Propositional Functions
+
+Quantifiers can be used to turn a propositional function into a proposition much like numbers.
+
+$\forall x P(x)$ - We would need to specify some universe, but this is a proposition and not a propositional function
+
+$P(x)$ - This is a propositional function
+
+$P(1)$ - This is a proposition
+
+### Negation
+
+When negating a for each we negate the propositional function (not P(x)) and flip the for each to be there exists.
+
+The oppossite is also true for negating a there exists.
+
+Examples:
+
+For all - $\neg \forall x P(x) = \exists x \neg P(x)$
+
+There exists - $\neg \exists P(x) = \forall x \neg P(x)$
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -110,3 +110,5 @@ L19:
- [[MarkovInequality.md]]
L20:
- [[CentralLimitTheroem.md]]
+
+The rest of the lectures discuss inference. I am not planning to read this as it will be covered in my Elements textbook.
diff --git a/Universe.md b/Universe.md
@@ -0,0 +1,12 @@
+:discrete: :logic: :math:
+# Universe
+
+U 1.4.1
+
+## Notes
+
+**Definition:** The universe in math is the set of all objects that bear consideration.
+
+Often we state the universe as the variable U.
+
+See also [[UniversalSet.md]] for the same concept. I created this note because the term bears remembering and I forgot what I called the universal set in the domain of stats and probability.
diff --git a/index.md b/index.md
@@ -49,6 +49,3 @@ ML
- [ ] Mamba
- [ ] Transformer
- [ ] KAN
-
-Physics
- - [ ]
