note update

dm

content/Discrete Math/Set.md

@@ -6,7 +6,7 @@ Author Profile:
 tags:
   - discrete_math
 Creation Date: 2023-12-18, 10:33
-Last Date: 2023-12-18T10:42:50+08:00
+Last Date: 2023-12-20T10:36:56+08:00
 References: 
 ---
 ## Abstract
@@ -20,7 +20,7 @@ $$
 
 
 
-## Notation
+## Notations
 ---
 ### Set-Roster Notation
 - A [[Discrete Math/Set|Set]] may be specified by writing all of its [[#Object]] between braces 
@@ -34,16 +34,67 @@ $$
 $$
 \{1,2,3, \ldots\}
 $$
+### Set-Builder Notation
+$$
+\{x \in U : P(x)\}
+$$
+$$
+\{x \in U | P(x)\}
+$$
+- The set of all `x` in `U` such that `P(x)`` is true
 
+### Replacement Notation
+$$
+\{t(x) : x \in A\}
+$$
+$$
+\{t(x) | x \in A\}
+$$
+- For elements x in `A`, we apply the function `t(x)`
+## Properties
+---
 ### Membership of a Set
+$$
+\in, \not\in
+$$
+$$
+b \in {a, b, c}
+$$
 
 ### Cardinality of a Set
-
+$$
+|\{a, b, c\}| = 3
+$$
+- The size of the [[Discrete Math/Set|Set]]
 
 
 
 ## Terminologies
 ---
 ### Object
 - Members or elements of [[Discrete Math/Set|Set]]
-- Example: `1`, `2`, `3` are objects in the set of [[Integer (整数)]]
+- Example: `1`, `2`, `3` are objects in the set of [[Integer (整数)]]
+
+### Subset
+$$
+A \subseteq B
+$$
+iff
+$$
+\forall x (x \in A \land x \in B)
+$$
+### Superset
+$$
+B \supseteq A
+$$
+- `B` is the supper `A`
+
+
+### Proper Subset
+$$
+A \subseteq B
+$$
+iff
+$$
+(A \subseteq B) \land (A \ne B)
+$$