commit cbb41d56fd32d89cc0f9e920150039251d5fb5ed
parent d025a46e14c753214cc04a6bba5663aaa219dc7a
Author: Andrew <andrewlaack1@gmail.com>
Date: Mon, 28 Oct 2024 09:10:50 -0500
Took discrete math notes
Diffstat:
5 files changed, 44 insertions(+), 6 deletions(-)
diff --git a/Closure.md b/Closure.md
@@ -1,4 +1,4 @@
-:lin-alg:
+:lin-alg: :discrete:
# Closure
Khan
@@ -8,3 +8,11 @@ Khan
**Definition:** Closure means that performing some arbitrary operation (pick one, but not necessarily all) on any member of a set will result in another element of a set.
In the context of subspaces, we have closure under scalar multiplication and vector addition because these operations on any element of the [[LinearSubspace.md]] set results in another element of the set (by definition).
+
+## Discrete Math
+
+**Definition:** A closure is when we have a property such that the relation contains the original set of the relation and has the minimum number of extra components to be closed with respect to the property.
+
+Closure under addition means we have the minimum number of elements in a relation such that the domain and codomain are included in the first and second components of ordered pairs, and all other elements necessary for addition to result in another element of the relation are included.
+
+An important note is that there are often many relations for which the relation has the property and contains the codomain and domain, but we are only interested in the smallest one, the one that has the fewest other elements in it.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -169,8 +169,10 @@ Unit 9.3 (Representing Relations)
- [Loop](Loop.md)
Unit 9.4 (Closures of Relations)
- - TransitiveClosure
- - ReflexiveClosure
- - SymmetricClosure
- - DiagonalRelation
- - Closure
+ - [TransitiveClosure](TransitiveClosure.md)
+ - [ReflexiveClosure](ReflexiveClosure.md)
+ - [SymmetricClosure](SymmetricClosure.md)
+ - [Closure](Closure.md)
+
+Unit 9.5 (Equivalence Relations)
+ - EquivalenceRelation
diff --git a/ReflexiveClosure.md b/ReflexiveClosure.md
@@ -0,0 +1,10 @@
+:discrete:
+# Reflexive Closure
+
+Ch 9.4
+
+## Notes
+
+**Definition:** A reflexive closure is a closure of a relation with respect to some property such that xRx for all x=x.
+
+When shown as a zero one matrix, this will manifest as the main diagonal being all 1's.
diff --git a/SymmetricClosure.md b/SymmetricClosure.md
@@ -0,0 +1,8 @@
+:discrete:
+# Symmetric Closure
+
+Ch 9.4
+
+## Notes
+
+**Definition:** A symmetric closure is the closure of some relation under some property such that if xRy then yRx.
diff --git a/TransitiveClosure.md b/TransitiveClosure.md
@@ -0,0 +1,10 @@
+:discrete:
+# Transitive Closure
+
+Ch 9.4
+
+## Notes
+
+**Definition:** A transitive closure is the closure of a relation under some property such that each element where there is a path from one to another is directly connected.
+
+This can be thought of as fully connecting any connected components.
