Technical Question

How do I connect mathematical terminology (e.g., mapping, function, number, sequence, and set), to the implementation of Python programs that declare and call functions and declare and manipulate variables?

Hello World with Highlighting

# declare multiple variables
hello = "hello"
world = "world"
space = " "
value = .50
message = world + space + hello
print(f"The message is: {message}")
print(f"The value is: {value}")

Using a mapper with a Sequence

def square(value: int):
    return value * value

def mapper(f, sequence):
    result = (  )
    for element in sequence:
        result += ( f(element), )
    return result

squared_range = mappper(square, range(10))
print(squared_range)

Understanding the Monoid

• A monoid is an ordered pair $(S, \otimes)$ for a set $S$ and any binary operator $\otimes$ that satisfies the following conditions:

  • Type Preservation: $\forall s_1, s_2 \in S$, $s_1 \otimes s_2 \in S$

  • Associative Property: $\forall s_1, s_2, s_3 \in S$, $(s_1 \otimes s_2) \otimes s_3 = s_1 \otimes (s_2 \otimes s_3)$

  • Identity Element: $\exists \epsilon \in S$, such that $\forall s \in S, \epsilon \otimes s = s$ and $s \otimes \epsilon = s$

• We often say that $S$ is a monoid under $\otimes$ with identity $\epsilon$

• If this is confusing, a monoid is a generalization of strings and integers!

• If you know how strings behave in Python or Java then you understand the monoid — monoid describes string-like structures!

Average Computation with Multisets

$$O = ((o_1, \ldots, o_n))$$

$$S = \sum_{o_i \in O} o_i$$

$$A = \frac{S}{|O|}$$

What is the meaning of $o_i \in O$?

Where does this exist in Python code?

Explore the use of the sum function in Python!

Average Computation with Multisets

$$O = ((o_1, \ldots, o_n))$$

$$S = \sum_{o_i \in O} o_i$$

$$A = \frac{S}{|O|}$$

What is the meaning of $o_i \in O$?

Where does this exist in Python code?

Explore the use of the sum function in Python!

Summary of the Abstraction Jumping

• What is the connection between the discrete mathematical structures and the Python programs?

• Connections between discrete mathematics and Python

  • Generic file: a sequence of sequences

  • Names in the file: a set of strings

  • Emails in the file: a set of ordered pairs forming a relation

  • Temperatures in the file: a multiset of integers

• When might the emails in the file be a mapping? When might the temperatures in the file be a sequence?

Simpler Slide with Bulleted List

• Item 1

  • Sub list
  • Sub list again

• Item 2

• Item 3

• Item 4

Sample Diagrams in Mermaid

Additional Diagram Slide

Separate Diagram Slide Again

Separate Diagram Slide Last