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