Exploring Mathematics and Data Science with Technology

(a)

Sets may be declared in Python and SageMath using the syntax {2,8,1,4}. Use this syntax and the equality operator == to confirm that \(\{100,101,100\}\text{,}\) \(\{101,100\}\text{,}\) and \(\{100,101\}\) are all the same set (since they all contain exactly the same numbers).

(b)

The union of many sets describes anything that belongs to any of the given sets. For example, the union of \(\{1,2,3\}\) with \(\{3,4,5\}\) may be found using the code {1,2,3} | {3,4,5}, resulting in \(\{1,2,3,4,5\}\text{.}\)

Find the union of the three sets \(\{1,2,4,7\}\text{,}\) \(\{2,5,7,9\}\text{,}\) and \(\{0,2,3,7\}\text{.}\)

(c)

The intersection of many sets describes only the things that belong to all of the given sets. For example, the intersection of \(\{1,2,3\}\) with \(\{3,4,5\}\) may be found using the code {1,2,3} & {3,4,5}, resulting in \(\{3\}\text{.}\)

Find the intersection of the three sets \(\{1,2,4,7\}\text{,}\) \(\{2,5,7,9\}\text{,}\) and \(\{0,2,3,7\}\text{.}\)

(d)

Sets may contain other kinds of data, such as strings.

Find the union and intersection of the sets {1,2,3} and {1,"2","three"}.

(a)

Lists use hard brackets [2,8,1,4] instead of curly brackets. Use == to verify that each of the lists [100,101,100], [101,100], and [100,101] are not considered equal to each other, since even though they contain the same data, it appears in different orders and frequencies.

(b)

The concatentation of many lists attaches them in order from end-to-end. For example, the concatenation of [1,2,3] with [3,4,5] may be found using the code [1,2,3] + [3,4,5], resulting in [1,2,3,3,4,5]. (Note that 3 is repeated since lists allow repeats.)

Find the concatenation of the three lists \([1,2,4,7]\text{,}\) \([2,5,7,9]\text{,}\) and \([0,2,3,7]\text{.}\)

(c)

Like sets, lists may contain data besides raw numbers, such as strings.

Find the concatenation of the lists [1,2,3] and [1,"2","three"].

(d)

Elements of a list can be changed as shown in 2.2.1.

favorite_numbers = [1,"two",7,3.14]
# Characters after the hashtag sign are comments and don't affect code.
# Python indexes with counting nunmbers, starting with 0 (known as 0-indexing).
favorite_numbers[0] = "One."  # changes the initial element 1 to "One."
favorite_numbers[2] = 42      # changes the element 7 to 42
print(favorite_numbers)       # outputs ['One.', 'two', 42, 3.14]

Let favorite_things = ["raindrops", "roses", "whiskers", "kittens"], and then use the techniques from 2.2.1 to change “roses” and “kittens” into some of your favorite things (leaving "raindrops" and "whiskers" in place).

(e)

Tuples are similar to lists in that they store ordered collections of data, but are used in different contexts. An important difference is that their elements are fixed. (That is, lists are mutable while tuples are immutable.)

Copy 2.2.2 to a Code cell, and replace the two indicated comments appropriately.

triple_list = ["one", "two", "three"]
triple_tuple = ("one", "two", "three")

# Attempts indented block of code that may cause an error
try:
    print("Trying to change 'two' to 2 in the list.")
    # REPLACE this line to change triple_list appropriately
# Handles possible error
except TypeError:
    print("This should never appear since list elements can be replaced.")

# Attempts indented block of code that may cause an error
try:
    print("Trying to change 'three' to 3 in the tuple.")
    # REPLACE this line to try to change triple_list appropriately
# Handles possible error
except TypeError:
    print("This should appear since tuple elements cannot be replaced.")

print(triple_list, triple_tuple) # outputs ['one', 2, 'three'] ('one', 'two', 'three')

(a)

The dictionary student_dict = {"name": "Jessica", "age": 19, "favorite_teacher": "Dr. Clontz"} represents the student data given in 2.2.3 as keys and values.

Create a dictionary with the same keys for someone in your group.

(b)

Values stored in a dictionary can be easy retrieved by looking them up by key. For example, student_dict["favorite_teacher"] equals "Dr. Clontz", and print(f"{student_dict['name']} is {student_dict['age']} years old.") outputs Jessica is 19 years old.. (Note the switching of single and double quotes.)

Create a code cell that outputs a sentence similar to the following, using the dictionary you created in the previous task.

[Student name]'s favorite teacher is [teacher name].

(a)

Using the range() tool, we can quickly create lists of counting numbers. For example, list(range(10)) returns the list [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Note that this list contains ten different elements, but doesn't contain the number 10 itself.

By providing two numbers, you can specify the starting number (included) and ending number (not included). For example, list(range(-5,5)) == [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4].

Use a range to create the list [-10, -9, ..., 9, 10].

(b)

Comprehension allows us to use formulas when defining a list. For example, [2*n for n in range(10)] == [0, 2, 4, ..., 18].

Use comprehension to square each of the numbers in the list [-10, -9, ..., 9, 10], producing [100, 81, ..., 81, 100].

(c)

Comprehension can also be used with sets and dictionaries. For example, {f"{n} doubled": 2*n for n in range(10)} creates the dictionary {'0 doubled': 0, '1 doubled' 2, ..., '9 doubled': 18}.

Use comprehension to create the dictionary {'-10 squared': 100, '-9 squared': 81, ..., '10 squared': 100}.