commit 158ab604569d6ef842da2399a6cfda2edd57aeff
parent 174a0fb861804b9ce74ad4b77df54ffad94985d6
Author: Andrew <andrewlaack1@gmail.com>
Date: Mon, 16 Sep 2024 09:21:21 -0500
Added yesterday's notes
Diffstat:
9 files changed, 74 insertions(+), 6 deletions(-)
diff --git a/Algorithms.md b/Algorithms.md
@@ -79,3 +79,4 @@ L6:
- [BekensteinBound](BekensteinBound.md)
- [OracleComputer](OracleComputer.md)
+- [Invariance](Invariance.md)
diff --git a/BinomialCoefficient.md b/BinomialCoefficient.md
@@ -7,6 +7,8 @@ L4
**Definition:** A binomial coefficient is represented by two numbers and has a singular evaluation. The evaluation describes the number of unique subsets of the length denoted by the bottom value that can be created given a set of the length denoted by the top value.
+The reason it is called the binomial coefficient is because it can be used in the expansion of binomials (ie. (x+y)^5). To use it in this case we multiply the applicable coefficient with the number of ways to select that number of a coefficient. This idea is also described as the binomial theorem.
+
### Formula
(n) = n! / ((r!(n-r)!)
diff --git a/Combination.md b/Combination.md
@@ -0,0 +1,10 @@
+:discrete:
+# Combination
+
+TB 6.3
+
+## Notes
+
+**Definition:** A combination is a unique selection of elements from a given set.
+
+The difference between a combination and a permutation is rearrangements of combinations are still considered the same whereas the opposite is true for permutations.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -125,11 +125,12 @@ Unit 6.2 (Pigeonhole principle)
- [RamseyNumbers](RamseyNumbers.md)
Unit 6.3 (Permutations and Combinations)
- - Permutation
- - RPermutation
- - Combination
- - RCombination
+ - [Permutation](Permutation.md)
+ - [RPermutation](RPermutation.md)
+ - [Combination](Combination.md)
+ - [RCombination](RCombination.md)
Unit 6.4 (Binomial Coefficient & Identities)
- - Binomial Coefficient (think about expansion of distribution as a counting problem)
- -
+ - [BinomialCoefficient](BinomialCoefficient.md)
+ - [PascalsIdentity](PascalsIdentity.md)
+ - [VandermondesIdentity](VandermondesIdentity.md)
diff --git a/Invariance.md b/Invariance.md
@@ -0,0 +1,10 @@
+:cs: :ml:
+# Invariance
+
+SS
+
+## Notes
+
+**Definition:** Invariance in ML describes changes to objects such that the model should still interpret the object the same way.
+
+There are a few different types including translational, rotational, and size invariance.
diff --git a/PascalsIdentity.md b/PascalsIdentity.md
@@ -0,0 +1,8 @@
+:discrete:
+# Pascals Identity
+
+Ch 6.4
+
+## Notes
+
+**Definition:** Pascal's identity is the idea that n+1 choose r is equivalent to n choose r plus n choose r-1.
diff --git a/RCombination.md b/RCombination.md
@@ -0,0 +1,12 @@
+:discrete:
+# r-Combination
+
+Ch 6.3
+
+## Notes
+
+**Definition:** An r-Combination is a combination of length r.
+
+The function to denote r-combinations of a set length n is C(r,n). There are other ways to state it, but I prefer this.
+
+$C(r,n) = \frac{n!}{r!(n-r)!}$
diff --git a/RPermutation.md b/RPermutation.md
@@ -0,0 +1,14 @@
+:discerte:
+# r-Permutations
+
+TB 6.3
+
+## Notes
+
+**Definition:** r-Permutations are permutations that have a lenght of r.
+
+An important functions is P(n, r) where this denotes the number of r-permutations of a set with a length of n.
+
+The formula for P(n,r) is as follows:
+
+$P(n,r) = \frac{n!}{(n-r)!}$.
diff --git a/VandermondesIdentity.md b/VandermondesIdentity.md
@@ -0,0 +1,10 @@
+:discrete:
+# Vandermonde's Identity
+
+Ch 6.4
+
+## Notes
+
+**Definition:** Vandermonde's identity is an identity that describes n+m choose k as a sum of all ways to select 0 of one and k of the other 1 of one and k-1 of the other and so on.
+
+$\binom{n+m}{k} = \sum^k_{i=0} \binom{n}{i} \binom{m}{k-i}$
