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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`
|
