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+[\<- 2.1](2.1.md)
+
+---
+
+# 2.2 Set Operations p1
+
+We can represent sets in a **Venn Diagram** as circles/ellipses that lie inside some **universal set**
+
+```
++----------+
+|          |
+|  +--+  u |
+|  | s|    |
+|  |  |    |
+|  +--+    |
++----------+
+```
+
+- "rectangle is universal set `u`
+
+- Ex.
+	- Let `u` = ℤ
+	- 'A' = `{2k | k∈ℤ}`
+	- 'B' = `{3l | l∈ℤ}`
+	1. Draw a Venn Diagram and write at least on element in each "region"
+
+```
++---------------------+
+|                     |
+|  +-------+      u   |
+|  | A     |          |
+|  |    +-------+     |
+|  |    |  |    |     |
+|  +----|--+    |     |
+|       |     B |     |
+|       +-------+     |
++---------------------+
+```
+
+- `u` is representing ℤ
+- Examples that belong in each region:
+	- Inside 'A', outside 'B'
+		- 2
+		- 4
+	- Inside 'B', outside 'A'
+		- 3
+		- -9
+	- Inside both 'A' and 'B'
+		- 6
+		- 0
+		- -18
+	- Outside of both 'A' and 'B' (but inside `u`)
+		- 7
+		- 5
+
+- **Definitions**: If 'A', 'B' are sets, the
+	1. **Intersection** of 'A' and 'B', written `A∩B`, is the set of all elements in 'A' **and** 'B'
+	2. **Union** of 'A' and 'B', written `A∪B`, is the set of all elements in 'A' **or** 'B'
+		- the "or" is inclusive, btw
+
+### Intersection Venn Diagram
+
+```
++---------------------+
+|    A↓               |
+|  +-------+      u   |
+|  |       |          |
+|  |    +-------+     |
+|  |    |██|    |     |
+|  +----|--+    |←    |
+|       |       |B    |
+|       +-------+     |
++---------------------+
+```
+
+- `A∩B` = shaded
+
+### Union Venn Diagram
+
+```
++---------------------+
+|    A↓               |
+|  +-------+      u   |
+|  |███████|          |
+|  |████+-------+     |
+|  |████|██|████|     |
+|  +----|--+████|←    |
+|       |███████|B    |
+|       +-------+     |
++---------------------+
+```
+
+- `A∪B` = shaded
+
+|Words|Logic|Sets|
+|-----|-----|----|
+|and  |^    |∩   |
+|or   |∨    |∪   |
+
+- Describe `A∩B`, `A∪B` in the earlier example
+	- `A∩B` = multiples of 2 **and** 3
+		- multiples of 6
+	- `A∪B` = multiples of 2 **or** 3
+		- 2, 3, 4, 6, 8, ...
+
+---
+
+- **Definition**: If `A∩B` = Ø, we say 'A' and 'B' are **disjoint**
+
+```
++---------------------+
+|                     |
+|  +-------+      u   |
+|  | A     |          |
+|  |       |          |
+|  |       |   +----+ |
+|  +-------+   | B  | |
+|              |    | |
+|              +----+ |
++---------------------+
+```
+
+- Notice how 'A' and 'B' don't overlap at all
+
+- **Definition**: Fix a universal set `u`, and `S ≺= u`. The **complement** of 'S', written s̅ or s^c, is the set of elements `u` not in 'S'
+
+**complement**
+
+```
++---------------------+
+|█████████████████████|
+|██+-------+██████u███|
+|██| S     |██████████|
+|██|       |██████████|
+|██|       |██████████|
+|██+-------+██████████|
+|█████████████████████|
+|█████████████████████|
++---------------------+
+```
+
+- Everything outside of 'S' and inside `u` is shaded
+	- the shaded region is s̅
+
+- **Definition**: The **difference** of 'A' and 'B', written `A-B` or `A\B`, is the set of elements **in 'A' but not 'B'**
+
+```
++---------------------+
+|    A↓               |
+|  +-------+      u   |
+|  |███████|          |
+|  |████+-------+     |
+|  |████|  |    |     |
+|  +----|--+    |←    |
+|       |       |B    |
+|       +-------+     |
++---------------------+
+```
+
+- The shaded region is `A-B`
+
+- **Note**: `A-B` = `A∩¯(B̄̄̄̄̄̄̄̄̄)`
+	- (line above 'B' -> complement of B)
+
+---
+
+## Set Identities
+
+| | | |
+|-|-|-|
+|**Commutative**|`A∪B` = `B∪A`|`A∩B` = `B∩A`|
+|**Associative**|`A∪(B∪C)` = `(A∪B)∪C`|`A∩(B∩C)` = `(A∩B)∩C`|
+|**De Morgan (for sets)|`¯(A∩B)` (not in ('A and 'B')) = `¯(A)∪¯(B)`|`¯(A∪B)` (not in ('A or 'B')) = `¯(A)∩¯(B)`|
+
+---
+
+# 2.2 Set Operations p2
+
+Ex: Let `A ≺= B`. Prove that `A∪B ≺= B`
+- Proof: Let x∈A∪B. Then, x∈A or x∈B.
+	- Case 1: If x∈A, then x∈B since `A ≺= B`
+	- Case 2: If x∈B, then we are done 
+	- ∴ x∈B which implies `A∪B ≺= B` ▣
+	
+- **Theorem**: Let 'A', 'B' be sets. 'A' = 'B' iff `A ≺= B` and `B ≺= A`