src/sorting_algorithms/sorting

complete the 4 blank sorting algorithms

to test them

1. run sorting_algorithms/testing_main.py to test your code
2. you can see plots of time used in plots, file name is Time used vs Lengths.png

src/sorting_algorithms/sorting

complete count sort and radix sort

to test them

1. run sorting_algorithms/testing_main.py to test your code
2. you can see plots of time used in plots, file name is Time used vs Lengths (Linear Sorts).png

src/data_structures/binary_search_tree and src/data_structures/heap

complete functions of the 2 data structures

to test them

1. install dependency graphviz by running pip install graphviz in command line
2. run data_structures/testing_main.py to test your code
3. you can see plots of time used in plots/bsts and plots/heaps

完成这次的作业后请把作业发到我的企业微信。前3次作业已完成的同学请把test通过的截图（或者代码截图）发到我的企业微信。谢谢。

Find the union, and intersection of:

a) ${ 1, 2, 3, 4,5}$ and ${−1, 1, 3, 5,7}$

b) ${ x ∈ \text{R} | x > 7}$ and ${ x ∈ \text{R} | x > 5}$

prove the universal properties of set union and intersection, for sets $A,B \subseteq D$ (D is the domain)

1. $\forall X\subseteq D. A \cup B \subseteq X \iff (A \subseteq X \text{ and } B \subseteq X)$
2. $\forall X\subseteq D, X\subseteq A \text{ and } X\subseteq B \iff X \subseteq (A\cap B)$

Hints:

• remember how to prove 'if and only if' (two steps)
• use the definitions of $\cup$ and $\cap$
  1. $\forall x, x\in A\cup B \iff x\in A \vee x\in B$
  2. $\forall x, x\in A\cap B \iff x\in A \wedge x\in B$
• use the definition of $\subseteq$
  • $A\subseteq B\iff \forall a \in A,a\in B$

prove the universal properties of big union and big intersection, for a family of sets $F \subseteq P(D)$ (D is the domain)

1. $\forall X \subseteq D. \bigcup F \subseteq X \iff (\forall S \in F, S \subseteq X)$
2. $\forall X \subseteq D. X \subseteq \bigcap F \iff (\forall S\in F, X\subseteq S)$

Hints:

• $F$ is a set of sets
• the proof is quite similar to the previous proof

Let A={1,2,3,4}, B={a,b,c,d} and C={x,y,z}. Let R = {(1,a),(2,d),(3,a),(3, b),(3,d)}: A → B and S = {(b, x),(b, y),(c, y),(d,z)}: B → C.

Draw the internal diagrams of the relations. What is the composition S ◦ R : A → C.

go to the online editor

query the table with columns: cutomer - item - status

NOTE: assume order_id is the same as shipping_id

result should look like: