---
# System prepended metadata

title: 'Ćwiczenia 3, grupa śr. 10-12, 15 marca 2023'
tags: [pwit, ćwiczenia, SYK23]

---

# Ćwiczenia 3, grupa śr. 10-12, 15 marca 2023

###### tags: `SYK23` `ćwiczenia` `pwit`

## Deklaracje

Gotowość rozwiązania zadania należy wyrazić poprzez postawienie X w odpowiedniej kolumnie! Jeśli pożądasz zreferować dane zadanie (co najwyżej jedno!) w trakcie dyskusji oznacz je znakiem ==X== na żółtym tle.

**UWAGA: Tabelkę wolno edytować tylko wtedy, gdy jest na zielonym tle!**


:::danger
|                        | 1     | 2   | 3   | 4   | 5   | 6   | 7   | 8   |
| ----------------------:| ----- | --- | --- | --- | --- | --- | --- | --- |
Mateusz Biłyk            |   X   |  X  |     |     |     |     |     |     | 
Mikołaj Dworszczak       |       |     |     |     |     |     |     |     |
Kacper Jóźwiak           |       |     |     |     |     |     |     |     |
Dominik Kiełbowicz       |       |     |     |     |     |     |     |     |
Michał Kolasa            |       |     |     |     |     |     |     |     |
Konrad Kowalczyk         |   X   |  X  |  X  |     |  X  |     |     |     |
Oskar Kropielnicki       |   X   |  X  |  X  |     |  X  |     |     |     |
Anna Krysta              |   X   |  X  |  X  |  X  |  X  |  X  |  X  |  X  |
Jakub Krzyżowski         |   X   |  X  |     |     |     |     |     |     |
Oskar Kubkowski          |   X   |  X  |  X  |  X  |  X  |     | ==X==   |     |
Patryk Maciąg            |       |     |     |     |     |     |     |     |
Mateusz Mazur            |  X     | X    |  X   |     |     |     |     |     |
Barbara Moczulska        |   X   |  X  |  X  |     |  X  |     |     |     |
Kacper Sojda             |   X    |  X   |  X   |  X   |  X   |     |  X   |     |
Marta Strzelec           |   x    |  x   |  x   |     | ==x==    |     |  x   |     |
Mikołaj Swoboda          |  X    | X   | X   |     | X   |     | X   |     |
Filip Szczepański        |   X    |     |     |     |     |     |     |     |
Julian Włodarek          |   X   |     |     |     |     |     |     |     |
Beata Łakoma             |    x   |  x   |  x   |  x   |  x   |     | x    |     |
Michał Łukasik           |   X   |  X  |  X  |     |  X  |     |     |     |
:::





:::info
**Uwaga:** Po rozwiązaniu zadania należy zmienić kolor nagłówka na zielony.
:::




## Zadanie 1
:::success
Autor:		Mateusz Biłyk	
:::
![](https://i.imgur.com/3aDJ5hU.png)

![](https://i.imgur.com/aio3qgn.png)

| nr  | wejscie                                   | wyjscie                                   |
| --- | ----------------------------------------- | ----------------------------------------- |
| 1   |                                           | (x,1)                                     |
| 2   | (x,1)                                     | (x,1),(y,2)                               |
| 3   | (x,1),(y,2)                               | (x,1),(y,2),(i,3)                         |
| 4   | (x,1),(y,2),(i,3),(t,5),(x,6),(y,7),(i,8) | (x,1),(y,2),(i,3),(t,5),(x,6),(y,7),(i,8) |
| 5   | (x,1),(y,2),(i,3),(t,5),(x,6),(y,7),(i,8) | (x,1),(y,2),(i,3),(t,5),(x,6),(y,7),(i,8) |
| 6   | (x,1),(y,2),(i,3),(t,5),(x,6),(y,7),(i,8) | (y,2),(i,3),(t,5),(x,6),(y,7),(i,8)       |
| 7   | (y,2),(i,3),(t,5),(x,6),(y,7),(i,8)       | (i,3),(t,5),(x,6),(y,7),(i,8)             |
| 8   | (i,3),(t,5),(x,6),(y,7),(i,8)             | (t,5),(x,6),(y,7),(i,8)                   |
| 9   | (x,1),(y,2),(i,3),(t,5),(x,6),(y,7),(i,8) | (x,1),(i,3),(t,5),(x,6),(i,8),(y,9)         
## Zadanie 2
:::success
Autor:		Mateusz Mazur	
:::

![](https://i.imgur.com/zk7DUrK.png)

![](https://i.imgur.com/EpYPglw.png)
$RD_{\circ}(1) = \{(x,?), (y,?), (z,?), (i,?), (t,?)\}$
$RD_{\bullet}(1) = RD_{\circ}(1) \setminus \{(x,?)\} \bigcup \{(x,1)\} =\{(x,1), (y,?), (z,?), (i,?), (t,?)\}$

$RD_{\circ}(2) = RD_{\bullet}(1)$
$RD_{\bullet}(2) = RD_{\circ}(2) \setminus \{(y,?)\}\ \bigcup \{(y,2)\} = \{(x,1), (y,2), (z,?), (i,?), (t,?)\}$

$RD_{\circ}(3) = RD_{\bullet}(2)$
$RD_{\bullet}(3) = RD_{\circ}(3) \setminus \{(i,?)\}\ \bigcup \{(i,3)\} = \{(x,1), (y,2), (z,?), (i,3), (t,?)\}$

$RD_{\circ}(4) = RD_{\bullet}(3) \bigcup RD_{\bullet}(8) = \{(x,1), (x,6), (y,2), (y,7), (z,?), (i,3), (i,8), (t,?) (t,5)\}$
$RD_{\bullet}(4) = RD_{\circ}(4)$

$RD_{\circ}(5) = RD_{\bullet}(4)$
$RD_{\bullet}(5) = RD_{\circ}(5) \setminus \{(t,?)\} \bigcup \{(t,5)\} = \{(x,1), (x,6), (y,2), (y,7), (z,?), (i,3), (i,8), (t,5)\}$ 

$RD_{\circ}(6) = RD_{\bullet}(5)$
$RD_{\bullet}(6) = RD_{\circ}(6) \setminus \{(x,1), (x,6)\} \bigcup \{(x,6)\} = \{(x,6), (y,2), (y,7), (z,?), (i,3), (i,8), (t,5)\}$ 

$RD_{\circ}(7) = RD_{\bullet}(6)$
$RD_{\bullet}(7) = RD_{\circ}(7) \setminus \{(y,7),(y,2)\} \bigcup \{(y,7)\} = \{(x,6), (y,7), (z,?), (i,3), (i,8), (t,5)\}$

$RD_{\circ}(8) = RD_{\bullet}(7)$
$RD_{\bullet}(8) = RD_{\circ}(8) \setminus \{(i,3),(i,8)\} \bigcup \{(i,8)\} = \{(x,6), (y,7), (z,?), (i,8), (t,5)\}$

$RD_{\circ}(9) = RD_{\bullet}(4)$
$RD_{\bullet}(9) = RD_{\circ}(9) \setminus \{(y,2),(y,7)\} \bigcup \{(y,9)\} = \{(x,1), (x,6), (y,9), (z,?), (i,3), (i,8), (t,?) (t,5)\}$

## Zadanie 3
:::success
Autor:	Oskar Kropielnicki		
:::

![](https://i.imgur.com/MXObpLR.png)

:::info
$RD_\bullet(1) = RD_\circ(1) \backslash \{(x, l) | l \in \text{Lab}\} \cup \{(x, 1)\}$
$RD_\bullet(2) = RD_\circ(2) \backslash \{(y, l) | l \in \text{Lab}\} \cup \{(y, 2)\}$
$RD_\bullet(3) = RD_\circ(3) \backslash \{(i, l) | l \in \text{Lab}\} \cup \{(i, 3)\}$
$RD_\bullet(4) = RD_\circ(4)$
$RD_\bullet(5) = RD_\circ(5) \backslash \{(t, l) | l \in \text{Lab}\} \cup \{(t, 5)\}$
$RD_\bullet(6) = RD_\circ(6) \backslash \{(x, l) | l \in \text{Lab}\} \cup \{(x, 6)\}$
$RD_\bullet(7) = RD_\circ(7) \backslash \{(y, l) | l \in \text{Lab}\} \cup \{(y, 7)\}$
$RD_\bullet(8) = RD_\circ(8) \backslash \{(i, l) | l \in \text{Lab}\} \cup \{(i, 8)\}$
$RD_\bullet(9) = RD_\circ(9) \backslash \{(y, l) | l \in \text{Lab}\} \cup \{(y, 9)\}$
<br>
$RD_\circ(1) = \{(v, ?) | v \in \{x, y, z, i, t\}\}$
$RD_\circ(2) = RD_\bullet(1)$
$RD_\circ(3) = RD_\bullet(2)$
$RD_\circ(4) = RD_\bullet(3) \cup RD_\bullet(8)$
$RD_\circ(5) = RD_\bullet(4)$
$RD_\circ(6) = RD_\bullet(5)$
$RD_\circ(7) = RD_\bullet(6)$
$RD_\circ(8) = RD_\bullet(7)$
$RD_\circ(9) = RD_\bullet(4)$
:::

### Iteracja 0.

$RD_\alpha(i) = \emptyset \quad (i \in \{1, 2, ..., 9\}; \alpha \in \{\circ, \bullet\})$

### Iteracja 1.

$\ast$ oznacza wartość, która w kolejnych iteracjach pozostaje taka sama.

$RD_\circ^\ast(1)   = \{(x, ?), (y, ?), (z, ?), (i, ?), (t, ?)\}$
$RD_\bullet^\ast(1) = \{(x, 1), (y, ?), (z, ?), (i, ?), (t, ?)\}$
$RD_\circ^\ast(2)   = \{(x, 1), (y, ?), (z, ?), (i, ?), (t, ?)\}$
$RD_\bullet^\ast(2) = \{(x, 1), (y, 2), (z, ?), (i, ?), (t, ?)\}$
$RD_\circ^\ast(3)   = \{(x, 1), (y, 2), (z, ?), (i, ?), (t, ?)\}$
$RD_\bullet^\ast(3) = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, ?)\}$
$RD_\circ(4)        = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, ?)\} \cup \emptyset$
$RD_\bullet(4)      = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, ?)\}$
$RD_\circ(5)        = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, ?)\}$
$RD_\bullet(5)      = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, 5)\}$
$RD_\circ(6)        = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, 5)\}$
$RD_\bullet(6)      = \{(x, 6), (y, 2), (z, ?), (i, 3), (t, 5)\}$
$RD_\circ(7)        = \{(x, 6), (y, 2), (z, ?), (i, 3), (t, 5)\}$
$RD_\bullet(7)      = \{(x, 6), (y, 7), (z, ?), (i, 3), (t, 5)\}$
$RD_\circ(8)        = \{(x, 6), (y, 7), (z, ?), (i, 3), (t, 5)\}$
$RD_\bullet^\ast(8) = \{(x, 6), (y, 7), (z, ?), (i, 8), (t, 5)\}$
$RD_\circ(9)        = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, ?)\}$
$RD_\bullet(9)      = \{(x, 1), (y, 9), (z, ?), (i, 3), (t, ?)\}$

### Iteracja 2.

$RD_\circ^\ast(4)   = \{(x, 1), (y, 2), (z, ?), (i, 3), (t, ?)\} \cup \{(x, 6), (y, 7), (z, ?), (i, 8), (t, 5)\} \\ \qquad = \{(x, 1), (x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, ?), (t, 5)\}$
$RD_\bullet^\ast(4) = \{(x, 1), (x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, ?), (t, 5)\}$
$RD_\circ^\ast(5)   = \{(x, 1), (x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, ?), (t, 5)\}$
$RD_\bullet^\ast(5) = \{(x, 1), (x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, 5)\}$
$RD_\circ^\ast(6)   = \{(x, 1), (x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, 5)\}$
$RD_\bullet^\ast(6) = \{(x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, 5)\}$
$RD_\circ^\ast(7)   = \{(x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, 5)\}$
$RD_\bullet^\ast(7) = \{(x, 6), (y, 7), (z, ?), (i, 3), (i, 8), (t, 5)\}$
$RD_\circ^\ast(8)   = \{(x, 6), (y, 7), (z, ?), (i, 3), (i, 8), (t, 5)\}$
$RD_\circ^\ast(9)   = \{(x, 1), (x, 6), (y, 2), (y, 7), (z, ?), (i, 3), (i, 8), (t, ?), (t, 5)\}$
$RD_\bullet^\ast(9) = \{(x, 1), (x, 6), (y, 9), (z, ?), (i, 3), (i, 8), (t, ?), (t, 5)\}$

## Zadanie 4
:::success
Autor:		Kacper Sojda	
:::

![](https://i.imgur.com/Bt5VItE.png)
![](https://i.imgur.com/XBIQ5Hr.png)

## Zadanie 5
:::success
Autor:		Marta Strzelec	
:::

![](https://i.imgur.com/lQWU50O.png)
![](https://i.imgur.com/lVU8I18.jpg)

## Zadanie 6
:::success
Autor:		Anna Krysta	
:::
![](https://i.imgur.com/tqQ10nL.jpg)
![](https://i.imgur.com/rY1fuQF.jpg)
![](https://i.imgur.com/xaznNMc.jpg)
![](https://i.imgur.com/Y09hmid.jpg)
![](https://i.imgur.com/vfOb0xy.jpg)



## Zadanie 7
:::success
Autor:	Oskar Kubkowski		
:::
![](https://i.imgur.com/EQrgVRg.png)
![](https://i.imgur.com/CKQca0l.png)
![](https://i.imgur.com/GwzCPNU.png)
![](https://i.imgur.com/AENa6hj.png)

| $l$ | $RD_{\circ} (l)$                                                         | $RD_{\bullet} (l)$                                                       |
| --- | ------------------------------------------------------------------------ | ------------------------------------------------------------------------ |
| 1   | {(x, ?), (y, ?), (t, ?), (i, ?), (z, ?)}                                 | {(x, 1), (y, ?), (t, ?), (i, ?), (z, ?)}                                 |
| 2   | {(x, 1), (y, ?), (t, ?), (i, ?), (z, ?)}                                 | {(x, 1), (y, 2), (t, ?), (i, ?), (z, ?)}                                 |
| 3   | {(x, 1), (y, 2), (t, ?), (i, ?), (z, ?)}                                 | {(x, 1), (y, 2), (t, ?), (i, 3), (z, ?)}                                 |
| 4   | {(x, 1), (y, 2), (t, ?), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)} | {(x, 1), (y, 2), (t, ?), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)} |
| 5   | {(x, 1), (y, 2), (t, ?), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)} | {(x, 1), (y, 2), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}         |
| 6   | {(x, 1), (y, 2), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}         | { (y, 2), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}                |
| 7   | {(y, 2), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}                 | { (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}                        |
| 8   | {(i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}                         | {(z, ?), (t, 5), (x, 6), (y, 7), (i, 8)}                                 |
| 9   | {(x, 1), (y, 2), (t, ?), (i, 3), (z, ?), (t, 5), (x, 6), (y, 7), (i, 8)} | {(x, 1), (t, ?), (i, 3), (z, ?), (t, 5), (x, 6), (i, 8), (y, 9)}         |

| $ud(x,l)$ | $x$           | $y$           | $t$           | $i$           | $z$           |
| ------- | ----------- | ----------- | ----------- | ----------- | ----------- |
| $1$       | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ |
| $2$       | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ |
| $3$       | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ |
| $4$      | $\emptyset$ | $\emptyset$ | $\emptyset$ | {3,8}       | {?}         |
| $5$       | {1,6}       | {2,7}       | $\emptyset$ | $\emptyset$ | $\emptyset$ |
| $6$       | $\emptyset$ | {2,7}       | $\emptyset$ | $\emptyset$ | $\emptyset$ |
| $7$      | $\emptyset$ | $\emptyset$ | {5}         | $\emptyset$ | $\emptyset$ |
| $8$       | $\emptyset$ | $\emptyset$ | $\emptyset$ | {3,8}       | $\emptyset$ |
| $9$       | {1,6}       | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ |

## Zadanie 8
:::success
Autor:	Anna Krysta		
:::
![](https://i.imgur.com/OBSEtxR.jpg)
![](https://i.imgur.com/CQ6S1aG.jpg)
![](https://i.imgur.com/9gTjWxi.jpg)
![](https://i.imgur.com/uwqcgoL.jpg)
![](https://i.imgur.com/ZWSpzsw.jpg)
