functions

math/discrete/README.MD

@@ -3,7 +3,7 @@ layout: page
 title: Discrete Math
 permalink: /math/discrete/
 description: Discrete Math course and tips
-date: '2024-3-19'
+date: '2024-3-20'
 categories: Discrete
 tags:
 - Discrete
@@ -41,7 +41,6 @@ tags:
 
         * A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
 
-
 - Subsets: 
 
     * A ⊆ A ∪ B = B ∪ A
@@ -78,7 +77,7 @@ tags:
 
  * A ∈ P(A), ∅ ∈ P(A)
 
- * `If |A| = n, then |P(A)|` = <sup>n</sup>
+ * `If |A| = n, then |P(A)|` = 2<sup>n</sup>
 
  * `|P(A)|` = 2<sup>|A|</sup>
 
@@ -143,25 +142,63 @@ tags:
 - Important relations:
 
     1. **Reflexive:**
-    - Shorthand: \( R \subseteq \{(a, a) \mid a \in A\} \)
+    - Shorthand: \( R ⊆ \{(a, a) \| a ∈ A\} \)
     - Meaning: Every element is related to itself. Ia ⊆ R
+        - for all a ∈ A, aRa holds
 
     2. **Transitive:**
-    - Shorthand: \( R \circ R \subseteq R \)
-    - Meaning: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
+    - Shorthand: \( R ∘ R ⊆ R \)
+    - Meaning: If \( (a, b) ∈ R \) and \( (b, c) ∈ R \), then \( (a, c) ∈ R \).
+        - (aRb and bRc) -> aRc
 
     3. **Symmetric:**
-    - Shorthand: \( R = R^{-1} \)
-    - Meaning: If \( (a, b) \in R \), then \( (b, a) \in R \).
+    - Shorthand: \( R = R⁻¹ \)
+    - Meaning: If \( (a, b) ∈ R \), then \( (b, a) ∈ R \).
+        - When aRb <=> bRa
 
     4. **Antisymmetric:**
-    - Shorthand: \( R \cap R^{-1} \subseteq \{(a, a) \mid a \in A\} \)
-    - Meaning: If \( (a, b) \in R \) and \( (b, a) \in R \), then \( a = b \).
+    - Shorthand: \( R ∩ R⁻¹ ⊆ \{(a, a) \| a ∈ A\} \)
+    - Meaning: If \( (a, b) ∈ R \) and \( (b, a) ∈ R \), then \( a = b \).
+        - (aRb and bRa) -> (a = b)
     - This does not mean not-symmetric 
 
- - Equivalence relation is one where 1,2,3 all hold
+- Equivalence relation
+
+    -  is one where 1,2,3 all hold
+
+- Order relation 
+
+    - Partial order IFF 1,2,4 all hold
+        * clear hasse diagram can be drawn 
+            - items for which the relation doesn't hold will be drawn but not connected to the others in the diagram
+
+    - Total/Linear order:
+        * Partial order holds
+        * Totality: For any \( a, b ∈ A \), either \( (a, b) ∈ R \) or \( (b, a) ∈ R \).
+         - In other words: For any two distinct elements a and b, either a is related to b (a ≤ b), or b is related to a (b ≤ a).
+        * hasse diagram would be a straight line (all elements relate to one another in this set)
+
+    - Terms:
+        * Minimal: An element a is minimal if there is no b such that b precedes a.
+            -  Elements with nothing less than them (no predacessors)
+        * Minimum: An element a is a minimum if for all b, a precedes b.
+            -  Element that is less than everything else (either a set has 1 minimum or no minimum element)
+        * Maximal: An element a is maximal if there is no b such that a precedes b.
+            - follows from minimal
+        * Maximum: An element a is a maximum if for all b, b precedes a.
+            - follows from minimum
+
+#### Functions
+
+| Let f,g be two functions, (f:A -> B) , (g:B -> A) |                         |               |                  |                        |                                                                                 |                                                                                                        |
+| ------------------------------------------------- | ----------------------- | ------------- | ---------------- | ---------------------- | ------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------ |
+| Function f                                        | Horizontal line test    | Clasification | Invertibility    | G is \ \_ inverse of f | Definition                                                                      |                                                                                                        |
+| Onto                                              | Hits at least 1 point   | Surjective    | Right invertible | f ∘ g = Ib             | {f(a) | a ∈ A} = B every element in range (B) has a source                      | A function that maps one or more elements of A to the same element of B                                |
+| One to One                                        | Hits at most 1 point    | Injective     | Left Invertible  | g ∘ f = Ia             | if a1 != a2 then f(a1) != f(a2) or contrapositive if f(a1) = f(a2) then a1 = a2 | A function that always maps the distinct element of its domain to the distinct element of its codomain |
+| Onto and One to One                               | Hits exactly at 1 point | Bijective     | Invertible       | g ∘ f = Ia, f ∘ g = Ib | f⁻¹ = g                                                                         | A function that is both injective and surjective                                                       |
+| Identity Ia                                       | Hits exactly at 1 point | Bijective     | Invertible       | f ∘ Ia = f = Ib ∘ f    | f(a) = a                                                                        |
+
 
- - Order relation is one where 1,2,4 all hold 
 
 #### Links