fixed live pull issue
This commit is contained in:
venus
@@ -21,7 +21,8 @@ def index():
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@app.route("/api/vault-update") #webhook for vault updated
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@app.route("/api/vault-update") #webhook for vault updated
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def update_vault():
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def update_vault():
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# TODO SECURE THIS WITH SECRETTTTT or auth header
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# TODO SECURE THIS WITH SECRETTTTT or auth header
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print(build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR))# initialize the public notes from the private repo
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build.obsidian_vault(PRIVATE_VAULT_DIR) # initialize the private obsidian repo
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build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR)# initialize the public notes from the private repo
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return "vault-rebuilt"
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return "vault-rebuilt"
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@app.route ("/<filename>") # renders a filename if not otherwise specified
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@app.route ("/<filename>") # renders a filename if not otherwise specified
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def render_post(filename):
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def render_post(filename):
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79
content/Homework.md
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79
content/Homework.md
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#public
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# Homework
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Questions (P7.)
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1. a
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1. A. True
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2. B. Order does not matter in sets
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2. MISSISSIPPI
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3.
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1. $\subseteq$
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2. $\in$
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3. $\subseteq$
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4. $\in$
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5. $\in$ x wrong $\emptyset$ is a $\subseteq$ of all sets
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6. $\subseteq$
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4. 9.
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1. a) $\{S_4, S_5, S_9\}$
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2. b) **??**
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3. c) quadrillion
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4. d)
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1. F
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2. T (if order does not matter)
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3. T
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4. F
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5. T
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6. T
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7. F
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8. F
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9. F
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10. T
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11. F
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12. F
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13. T
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14. F
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15. T
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16. T
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5. 10.
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1. $D_1=\{1\}, D_2=\{1,2\}, D_{10}=\{1,2,5\}$
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2. b)
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1. T
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2. F
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3. T
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4. T
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5. T?
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6. F
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7. T
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8. F
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9. F
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10. F
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11. F
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12. T
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3. c) $|D_{10}|=3$, $|D_{19}|=2$
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4. D) $|\mathcal{D}|=9$
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| Questions | Answer |
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| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------- |
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| 1. <br> 1. (a) True or false? {red, white, blue} = {white, blue, red}. <br> 2. (b) What is wrong with this statement: Red is the first element of the set {red, white, blue}? | a. True<br>b. Order does not matter in sets |
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| 2. Which has the larger cardinality? The set of letters in the word MISSISSIPPI or the set of letters in the word FLORIDA ? | MISSIPPI |
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| 3. Fill in the blank with the appropriate symbol, ∈ or ⊆. <br> 1. (a) {1, 2, 3} {1, 2, 3, 4} <br> 2. (b) 3 {1, 2, 3, 4} <br> 3. (c) {3} {1, 2, 3, 4} <br> 4. (d) {𝑎} {{𝑎}, {𝑏}, {𝑎, 𝑏}} <br> 5. (e) ∅ {{𝑎}, {𝑏}, {𝑎, 𝑏}} <br> 6. (f) {{𝑎}, {𝑏}} {{𝑎}, {𝑏}, {𝑎, 𝑏}} | |
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| 9. Let 𝑆1 = {𝑜, 𝑛, 𝑒}, 𝑆2 = {𝑡, 𝑤, 𝑜}, 𝑆3 = {𝑡, ℎ, 𝑟, 𝑒, 𝑒}, and so on. <br> 1. (a) Find all 𝑘 ∈ {1, 2, . . . , 10} with \|𝑆𝑘\| = 4. <br> 2. (b) Find distinct indices 𝑗, 𝑘 ∈ ℕ with 𝑆𝑗 = 𝑆𝑘. <br> 3. (c) Find the smallest value of 𝑘 ∈ ℕ with 𝑎 ∈ 𝑆𝑘. <br> 4. (d) Let 𝒮 = {𝑆𝑘}40 𝑘=1. Determine whether the following statements are true or false. <br> 1. (i) 𝑆13 = {𝑛, 𝑒, 𝑖, 𝑡, ℎ, 𝑒, 𝑟} <br> 2. (ii) {𝑛, 𝑒, 𝑡} ⊆ 𝑆20 <br> 3. (iii) 𝑆1 ∈ 𝒮 <br> 4. (iv) 𝑆3 ⊆ 𝒮 <br> 5. (v) ∅ ∈ 𝒮 <br> 6. (vi) ∅ ⊂ 𝒮 <br> 7. (vii) ∅ ⊆ 𝒮 <br> 8. (viii) 𝑆1 ⊆ 𝑆11 <br> 9. (ix) 𝑆1 ⊆ 𝑆21 <br> 10. (x) 𝑆1 ⊂ 𝑆21<br> 11. (xi) {𝑛, 𝑖, 𝑒} ∈ 𝒮 <br> 12. (xii) {{𝑓, 𝑜, 𝑢, 𝑟}} ⊆ 𝒮 <br> 13. (xiii) 𝑢 ∈ 𝑆40 <br> 14. (xiv) 𝒫(𝑆9) ⊆ 𝒫(𝑆19) <br> 15. (xv) {𝑠, 𝑖} ∈ 𝒫(𝑆6) <br> 16. (xvi) 𝑤 ∈ 𝒫(𝑆2) | |
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| 10. For 𝑘 ∈ {1, 2, . . . , 20}, let 𝐷𝑘 = {𝑥 ∣ 𝑥 is a prime number which divides 𝑘} and let 𝒟 = {𝐷𝑘 ∣ 𝑘 ∈ {1, 2, . . . , 20}}.<br> 1. (a) Find 𝐷1, 𝐷2, 𝐷10, and 𝐷20. <br> 2. (b) True or False: <br> 1. (i) 𝐷2 ⊂ 𝐷10 <br> 2. (ii) 𝐷7 ⊆ 𝐷10 <br> 3. (iii) 𝐷10 ⊂ 𝐷20 <br> 4. (iv) ∅ ∈ 𝒟 <br> 5. (v) ∅ ⊂ 𝒟 <br> 6. (vi) 5 ∈ 𝒟 <br> 7. (vii) {5} ∈ 𝒟 <br> 8. (viii) {4, 5} ∈ 𝒟 <br> 9. (ix) {{3}} ⊆ 𝒟 <br> 10. (x) 𝒫(𝐷9) ⊆ 𝒫(𝐷6) <br> 11. (xi) 𝒫({3, 4}) ⊆ 𝒟 <br> 12. (xii) {2, 3} ∈ 𝒫(𝐷12) <br> 3. (c) Find \|𝐷10\| and \|𝐷19\|. <br> 4. (d) Find \|𝒟\| | |
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# Unit 1
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## Krish Hw 2
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1. A = {1, 2, 3}, B = {2, 3, 4} and C = {1, 2, 4}. List the elements of the specified set
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1. (a) A ∩ B; | {2,3}
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2. (b) A ∪ B; | {1,2,3,4}
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3. (c) C\A; | {1}
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4. (d) A ∪ (B ∩ C); | {2,1,3}
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5. (e) (A ∩ C) ∪ (B ∩ C); | {1,2,4}
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6. (f) A × B; | {(1,2),(1,3),(1,4),(2,3),(2,2),(2,4),(3,2),(3,3),(3,4)}
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7. (g) B × A; | {(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(4,1),(4,2) ,(4,3)}
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8. (h) (A × B) ∩ (B × A). | {(2,2),(3,3)}
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2. $M_2$ = {2, 4, 6, 8, 10, · · · } and $M_3$ = {3, 6, 9, 12, 15, · · · }. Find:
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1. (a) $M_2$ ∩ $M_3$;
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1. $M_6$
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2. (b) $M_3 \backslash M_2$
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1. $\{x|x=6k-3 \space\forall\space k \in\mathbb{N}\}$
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157
content/Unit 1 Sets and Logic.md
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157
content/Unit 1 Sets and Logic.md
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---
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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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Reference in New Issue
Block a user