Due date: 2026-01-15 23:59.

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This homework is designed to help you assess and refresh some of the background knowledge and notation that will be important for the rest of the course. Completing this assignment will help identify any areas that may need review and set you up for success in the rest of the course.

A set is a collection of objects that are referred to as elements (or members).

Usually curly brackets are used when listing out the elements of a set. The order in
which the elements appear is not important. For example,

The following symbols are used to indicate when an object is or is not an element of a set.

For example, we can make the following statements about the set X defined earlier.

Sets are often represented with capital letters, while their elements are often represented with lowercase letters, such as

A set that contains no elements is referred to as the empty set, and is denoted by
a special symbol.

A set is not required to consist of numbers. For example, one could have a set of
strings.

There are some additional restrictions on what collections of objects are allowed to
be sets. Those restrictions are out-of-scope for this class, but those who are interested
in why those restrictions exist should look into Russell's Paradox.

Consider the following three sets.

Each element of X is also an element of Y . This means that X is a subset of Y and the set Y is a superset of X. The following four symbols are used to indicate subset and superset relationships.

It is possible for a set to be both a subset and a superset of another set. For example,
each element of Z is an element of X and each element of X is an element of Z.
Therefore both X ⊆ Z and X ⊇ Z are true statements.

The set consisting of elements that are elements of both set X and set Y are referred to as
the intersection of X and Y . The set consisting of elements that are elements of set X, elements of set Y , or elements of both are referred to as the union of X and Y .

Suppose that X and Y are defined as follows.

In that case, the intersection of the sets is:

And the union of the sets is:

The intersection and union can be easily visualized with a Venn diagram.

Sets are not required to contain a finite number of elements. Four infinite sets are particularly useful and each of them is represented by a particular "blackboard bold" letter.

The natural numbers are the positive whole numbers starting from 1 (some definitions start from 0 instead). These are the numbers you would use if you were counting a set of objects out loud. The set of natural numbers is denoted with a blackboard bold ℕ.

The integers consists of the natural numbers, along with zero and the negative whole numbers. The set of integers is denoted with a blackboard bold ℤ.

The rational numbers consists of numbers that can be written as fractions of integers. The set of rational numbers is denoted with a blackboard bold ℚ.

The rational numbers do not include numbers that cannot be represented as fractions of integers.

The set that includes "all" of the numbers is called the real numbers, represented with a blackboard bold ℝ.

Interval subsets of the real numbers are denoted with parentheses and square brackets. A parenthesis is used when the endpoint is not included in the interval, and a square bracket is used when the endpoint is included.

It is acceptable to use a parenthesis on one side of the interval and a square bracket on the other side. The following intervals are perfectly valid.

A rigorous definition of the real numbers is outside of the scope of this class and it might seem like there is not much of a difference between the rational numbers and the real numbers (after all, there are an infinite number of both between 0 and 1). However, the set of real numbers is significantly "larger" than the rational numbers. Those who are interested in the distinction should look into Cantor's Diagonalization Proof.

Sometimes it is useful to define a set that combines elements from two other sets. For example, consider a chessboard. It has a set of a set of columns.

And it has a set of rows.

Any specific square on the chessboard can be described by the combination of a
column and a row.

Sets whose elements are formed by choosing one element from each of a list of input sets are called Cartesian products. The cross operator is used to indicate that a Cartesian product is being formed from sets.

Usually the elements of a set formed from a Cartesian product are written with parentheses.

For example, suppose we have the following two sets.

In that case, the Cartesian product of those two sets consists of the following elements.

Note that order within the parentheses is important for Cartesian products.

Cartesian products can be made from more than two input sets.

Cartesian products of the infinite sets we described earlier are particularly useful when describing something's position. For example, a car driving on the ground could have its position described by a pair of real numbers representing their longitude and latitude. An airplane could have its position described by a longitude, a latitude, and an altitude. (Let us assume that the car and airplane are not traveling so far that they have to worry about going around the entire planet.) Something that moves around on an infinite grid could have its position described by a pair of integers representing the x and y coordinates. Cartesian products between the same infinite set are often denoted with exponents.

A set can contain another set as an element. In these situations, it is very important to remember the differences between being an element of a set and of being a subset of a set. For example, consider the following two sets.

In this case,

The set of all subsets of a set X is referred to as the power set of X. This is usually denoted 2X or pow(X). For example,

Consider the following two sets

It is obvious from looking at those sets that for each element x in X, there is an element y in Y such that
y = 2x.

There is notation that can be used to write statements like this

For the X and Y sets above, we can make the following statement:

Which directly translates to "for each x in X, there exists a y in Y such that y=2x".

Listing all of the individual elements of a set is often not practical. Set builder notation can be used to quickly define the elements of a set based on a condition and a previously defined superset that the set elements are being drawn from. Set builder notation consists of the following pieces:

If we put together these pieces using the examples above, we can quickly define things like the set of even integers:

Or the set of points on a circle of radius 1 centered at the origin.