From 8bd507e9a494cee9de28f71f0f3d7e3397152da2 Mon Sep 17 00:00:00 2001
From: lshprung <lshprung@yahoo.com>
Date: Mon, 20 Apr 2020 11:25:56 -0700
Subject: Post-class 04/20

---
 2.1.md | 87 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 87 insertions(+)
 create mode 100644 2.1.md

(limited to '2.1.md')

diff --git a/2.1.md b/2.1.md
new file mode 100644
index 0000000..b09b8b9
--- /dev/null
+++ b/2.1.md
@@ -0,0 +1,87 @@
+[\<- 1.8](1.8.md)
+
+---
+
+# 2.1 Sets p1
+
+- A **Set** consists of elements, and it is **unordered**
+	- "x∈S" means `x` is an element of 'S'
+
+	- Note: {a,b,c} = {c,b,a} (order doesn't matter)
+	- Note: {a,b,c} = {a,a,b,b,c,c} (number of occurences doesn't matter)
+
+- **Definition**: Let 'S', 'T' be sets. We say 'S' is a **subset** of 'T', and write `S <= T`, if each element `x` of 'S' is also an element of 'T'
+	- To prove `S ≺= T`:
+		1. Let `x` be an arbitrary element of 'S'
+		2. Show it follows that x∈T
+
+- **Definition**: The set with no elements is called the **empty set**, denoted by Ø or {}
+	- For any set 'S',
+		1. `Ø ≺= S` (Ø is a subset of 'S')
+		2. `S ≺= S` ('S' is a subset of 'S')
+
+- **Definition**: If a set 'S' has `n` elements with n∈ℤ, `n >= 0`, then 'S' is a **finite set**. We say the **cardinality** of 'S' is `n` and write `|S| = n`
+	- If 'S' is not a finite set, then 'S' is an **infinite set**
+	- `(S = Ø) <-> (|S| = 0)`
+
+---
+
+## Power Sets
+
+**Definition**: The **Power Set** of 'S' is the set of all subsets of 'S' and is denoted `Ꝕ(S)`
+
+- Ex. Let 'S' = {`a`, `b`, `c`}. Write `Ꝕ(S)` in set notation.
+	- `Ꝕ(s)` = 
+		- Ø
+		- {`a`}
+		- {`b`}
+		- {`c`}
+		- {`a`, `b`}
+		- {`a`, `c`}
+		- {`b`, `c`}
+		- {`a`, `b`, `c`}
+		- Notice: all the elements of `Ꝕ(S)` are sets themselves (i.e. {a}∈Ꝕ(S) instead of just `a`)
+	- `|Ꝕ(S)| = 8`
+	- Can create elements of `Ꝕ(S)` with **bit strings** (sequence of 0's and 1's)
+
+- Ex. `S = {a, b, c}` ... bit string of length 3 (e.g. 011)
+	- 1st digit:
+		- 1 if a∈subset
+		- 0 if !(a∈subset)
+	- 2nd digit:
+		- 1 if b∈subset
+		- 0 if !(b∈subset)
+	- 3rd digit:
+		- 1 if c∈subset
+		- 0 if !(c∈subset)
+	- **Possible Bit Strings**:
+		- 000
+			- corresponds to Ø
+		- 100
+			- corresponds to {`a`}
+		- 010
+		- 001
+		- 110
+		- 101
+		- 011
+			- corresponds to {`b`, `c`}
+		- 111
+		- There are 8 possible bit strings because `2^3` = 8
+			- This leads to a theorem
+
+- **Theorem**: If `|S| = n`, then `|Ꝕ(S)| = 2^n`
+
+- **Definition**: Let 'S', 'T' be sets. The **product** of 'S' and 'T' is the set `{(s, t) | s∈S, t∈T}`
+	- (`s`, `t`) = ordered pair
+	- e.g. Let 'S' = {`a`, `b`, `c`} and 'T' = {1, 2}. Then,
+		- `S × T = {(a, 1), (b, 1), (c, 1), (a, 2), (b, 2), (c, 2)}`
+			- Notice: `|S × T|` = `6` = `3 · 2` = `|S| · |T|`
+
+- **Theorem**: If `|S| = n` and `|T| = m`, then `|S × T| = nm`
+
+- **Note**: (ℝ × ℝ × ℝ) = `{(a,b,c) | a,b,c∈ℝ}` is denoted ℝ^3
+	- we can similarly define ℝ^n for `n>3`
+
+---
+
+[2.2 ->](2.2.md)
-- 
cgit