---
tags:
- Math-301
- Math
- School
- notes
- Spring-25
- public
Slides: "[[Wed 1-7.pdf]]"
Topic: Sets and Logic
Unit: 1.1 - 1.6
---
# 1.1 Basic definitions
A set is a collection of objects. The objects in a set are called its elements or members.
> Let A={a,b,c}.
> $a\in A$ -- means *a* is an element of *A*
> $d \notin A$ -- means *d* is not an element of *A*
**Def**. the [[Cardinality]] of a finite set `S`, demoted `|S|`. is the number of elements in `S`.
In the example, `|A| = 3`.
### Notation for some sets of numbers
> Natural numbers: $\mathbb{N}$ = {1,2,3,...}
> Whole Numbers: $\mathbb{W}$ = {0,1,2,3,...}
> The set of integers: $\mathbb{Z}$ = {...-3,-2,-1,0,1,2,3,} = {0,1,-1,2,-2,3,-3...}
> Rational Numbers: $\mathbb{Q}$ = $\{\frac{a}{b}\vert a,b\in\mathbb{Z},b\neq0\}$
> Set of real numbers: $\mathbb{R}$
---
## 2 subsets
- Let *A* and *B* be sets. We say that *B* is a [[subset]] of *A*, if every element of *B* is also an element of *A*, denoted $B\subseteq A$
- Two sets are [[equal]], denoted $A = B$, if $A\subseteq B$ and $B\subseteq A$
>Ex. Let *A* = $\{x\vert x\in\mathbb{Z}\hspace{7px}and\hspace{7px}0\lt x\lt6\}$.
>We see that $A=\{1,2,3,4,5\}$.
>Note: $1\in A, 4\in A$, but $6\notin A$.
>$\{1,3,5\}\subseteq A,\{2\}\subseteq A$, but $\{2\}\notin A$
>{2,4,6}$\subsetneq A$ since $6 \notin A$
>|A| = 5.
>Ex. $\mathbb{N}\subseteq \mathbb{W} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$
[[Proper subset]]: $B\subset A$ if $B\subseteq A$ and $B\neq A$
### Intervals of $\mathbb{R}$
> (a,b) = {x$\in \mathbb{R}$ | a < x < b}, where $-\infty\le a \le b \le\infty$
> \[a,b] = {x$\in \mathbb{R}$ | a $\le$ x $\le$ b}, where $-\infty\lt a \le b \lt\infty$
> \[a,b) = \{x$\in \mathbb{R}$ | a $\le$ x < b}, where $-\infty\lt a \lt b \le\infty$
> \(a,b] = \{x$\in \mathbb{R}$ | a $\lt$ x $\le$ b}, where $-\infty\le a \lt b \lt\infty$
## 3. Collections of Sets
The elements of a set may themselves be sets, and so is is a collections of sets
> Ex. $\mathcal{C}$ = { {1}, {1, 2}, {1, 2, 3} }. Note that
> {1} $\in \mathcal{C}$, {1,2} $\notin\mathcal{C}$, {1,2,3} $\in\mathcal{C}$ **??**
> {1} $\subsetneq\mathcal{C}$ since $1\notin\mathcal{C}$
> {{1}, {1,2}} $\subseteq\mathcal{C}$
> {{1}, {1,2}} $\notin\mathcal{C}$.
### Indexed Collection of Sets
Let `I` be a set. Suppose $S_i$ is a set for each `i` $\in$ `I`.
Then we say that $\{S_i\}_{i\in I}$ = $\{S_i|i\in I\}$ Is called a [[collection of sets indexed]] by `I`.
> Ex. Let $S_n$ = (n-1, n) for each $n\in\mathbb{N}$. Then
> $\{S_n\}_{n=1}^{3}$ = {$S_1,S_2,S_3$} = $\{(0,1), (1,2), (2,3)\}$
> $\{S_n\}_{n\in\mathbb{N}}$ = $\{(0,1), (1,2), (2,3),...\}$
## 4. The Empty Set
Let `E` be a set with no elements. Then for any Set `A`, we have `E`$\subseteq$`A` [[(Vacuously True)]].
If `E'` is another set with no elements, then `E`$\subseteq$`E'` `E'`$\subseteq$`E`
So `E`=`E'` Therefore, there is a unique set with no elements. We call it the [[empty set]], denoted by $\emptyset$ = {}.
**Property: For every set `A`, $\emptyset\subseteq$`A`.**
## 5. The Power Set of a Set
the [[power set]] of a set `S` is the collection of all subsets of `S` and is denoted $\wp(S)$.
> $\wp(S) = \{A | A \subseteq S\}$.
>Ex. Let $A$ = $\{a,b,c\}$. Then
>$\wp(A)\{\emptyset\,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$
>Note: |$\wp(A)$ = 8 = $2^3$ = $2^{|A|}$.
## 6. Summarizing example
Consider the set:
> $S = \{1,2,\emptyset,\{a,b\}\}$. Then
> $2\in S, 2\subsetneq S$
> ${2}\notin S, 2\subseteq S$
> > $\emptyset, 2\subsetneq S$
# 1.2 Set Operations
## 1 Intersections and Unions
>Let `A` and `B` be sets. The [[Intersection]] of `A` and `B` is the set.
>$A\cap B$ {X | x $\in$ A and x $\in$ B}
>![[Pasted image 20260114205835.png]]
The [[Union]] of `A` and `B` is the set
> $A \cup B$ = {x | $\in$ a A or x $\in$ B } ## "Or" is the "Inclusive or"
> ![[Pasted image 20260114210548.png]]
> Ex. Let `A`={0,2,4,6,8}, `B`={0,3,6,9}, and C={1,2,3,4,5,6,8,9}. Then
> (a) $A\cap B$ = {0,6}
> (b) $A\cup B = \{0,2,4,6,8,3,9\}=\{0,2,3,4,6,8,9\}$
> (c) $B\cap C$ = {3}
> (d) $A\cup(B\cap C) = \{0,2,3,4,6,8\}$
> (e) $(A\cup B)\cap C = \{2,3,4\}$
> Note: $A\cup(B\cap C)\neq(A\cup B)\cap C$
**Def. We say that two sets A and B and [[disjoint]] if $A\cap B=\emptyset$**
![[Pasted image 20260115174934.png]]
> **Theorem. Let `A` and `B` be finite sets.**
> (a) $|A\cup B|=|A|+|B|-|A\cup B|$
> ![[Pasted image 20260115175142.png]]
> (b) if `A` and `B` and disjoint then $|A\cup B| = |A| + |B|$
> ![[Pasted image 20260115175132.png]]
> (c) If $A\subseteq B, then |A|\leq|B|$
## 2 Arbitrary Collections
Let $\mathcal{C}=\{S_i|i\in I\}$ be a collection of sets indexed by a set `I` (Assume $I\neq0$.) Then the [[Intersection]] of $\mathcal{C}$ is defined as
> $\cap\mathcal{C}=\cap_{i\in I} S_i=\{x|x\in S_i$ for all $i\in I\}$.
The [[Union]] of the collection $\mathcal{C}$ is the set
>$\cup\mathcal{C}=$ $\cup_{i\in I}S_i=\{x|x\in S_i$ for at least one $i\in I\}$
For a finite collection of sets indeed by $I=\{1,2 ... , n\}$
we often write the intersection and unions of $\mathcal{C}=\{S_1,S_2, ... ,S_n\}$ as
> $\cap_{i=1}^n S_i = S_1\cap S_2\cap S_3\cap ... \cap S_n$
> $\cup_{i\in I}^n S_i=S_1\cup S_2\cup ... \cup S_n$
Def. Let $\{\S_i|i\in I\}$ be an indexed collection of sets.
> (a) The collection is [[Mutually disjoint]] if for all $i,j\in I$, if $S_i\neq S_j, then S_I =\cap S_j\space\emptyset$
> Equivalently: for all $i,j\in I$, $S_i=S_j\space or\space S_i\cap S_j=\emptyset$
> (b) The collection is [[nested]] if for all $i,j\in I$, $S_i\subseteq S_j$ or $S_j\subseteq S_i$
$S=\{x\in\mathbb{W}|x\notin M_{5}, x\in M_3\}$
---
> $B_n$ = $\{x\in\mathbb{R}\space\vert\space |x| < n\}$
$B_1$ = (-1,1)
$B_2$ = (-2,2)
$B_3$ =(-3, 3)
> [!note] incluse vs exclusion
$[-1,1]\neq (-1,1)$ where $n=\mathbb{R}$
let $A$ = $(-1,1)$
let $B$ = $[-1,1]$
$A\subseteq B$
$A\neq B$
| Q | P | A |
| ------------------ | ---------------------- | ------------------ |
| Mutually disjoint? | $B_n\subseteq B_{n+1}$ | False |
| Nested?
| $B_n\subseteq B_{n+1}$ | True |
| Intersect | $B\cap B_n$ | $B_1$ |
| Union
| $B\cup B_n$ | $(-\infty,\infty)$ |