Sets

What Is a Mathematical Set?

A set is a well-defined collection of distinct objects. This is the foundational definition you need to know. Let's break down those key terms:

• Collection: This just means we are grouping things together. These things can be numbers, letters, names, or even other sets!
• Well-defined: This is crucial. It means we can determine with absolute certainty whether an object belongs to the collection or not. For example, "the collection of tall students in your class" is not well-defined because "tall" is subjective. However, "the collection of students in your class over 160cm tall" is well-defined because we can measure each student and get a definite yes or no answer.
• Distinct objects: Each object in a set must be unique. We don't list the same object more than once. The set of letters in the word "book" is $\{b,o,k\}$, not $\{b,o,o,k\}$.

The objects within a set are called its elements or members. We use curly braces $\{\}$ to enclose the elements of a set. For example, if we have a set named $A$ that contains the first five positive integers, we would write it as:

To show that an object is an element of a set, we use the symbol $\in$. To show it is not an element, we use $\notin$. Using our set $A$ above:

• $3\in A$ (Read as: "3 is an element of A")
• $7\notin A$ (Read as: "7 is not an element of A")

How Do We Describe and Write Sets?

There are two primary ways to describe the elements of a set. The method you choose often depends on which is clearer or more convenient for the specific set you're working with.

1. Roster Method (or Listing Method)

This is the most straightforward method. You simply list all the elements of the set, separated by commas, inside the curly braces. We've already seen this with $A=\{1,2,3,4,5\}$. The order in which you list the elements does not matter. The set $\{1,2,3\}$ is exactly the same as the set $\{3,1,2\}$.

For sets with many elements that follow a clear pattern, we can use an ellipsis ($...$) to save space. For example, the set of all integers from 1 to 100 can be written as:

2. Set-Builder Notation (or Rule Method)

This method is more powerful, especially for large or infinite sets. Instead of listing the elements, you describe them by stating a property or rule that its members must satisfy. The notation looks like this:

This is read as: "S is the set of all elements $x$ such that $x$ has a certain property." The vertical bar $\mid$ means "such that." A colon (:) is sometimes used instead.

For example, let's define the set $E$ of all even integers greater than 0. Listing them all is impossible! But with set-builder notation, it's easy:

Here's another example. The set $A=\{1,2,3,4,5\}$ can be written in set-builder notation as:

What Are the Different Types of Sets?

Sets can be categorized based on the number and nature of their elements. Understanding these types is essential for working with them.

• Finite Set: A set that has a limited, countable number of elements. The process of counting its elements would eventually come to an end. The set of days in a week, $\{\text{Monday, Tuesday, ... Sunday}\}$, is a finite set.
• Infinite Set: A set that has an unlimited number of elements. You could never finish counting them. The set of all whole numbers, $\{0,1,2,3,...\}$, is an infinite set.
• Empty Set (or Null Set): This is a special set that contains no elements at all. It is denoted by the symbol $\mathrm{\varnothing }$ or by empty curly braces $\{\}$. For example, the set of dogs that can fly is the empty set.
• Universal Set: This is the set of all possible elements relevant to a particular problem or context. It is denoted by the letter $U$. For example, if we are discussing sets of single-digit numbers, our universal set might be $U=\{0,1,2,3,4,5,6,7,8,9\}$. The universal set provides the boundaries for our discussion.

The number of distinct elements in a finite set is called its cardinality. The cardinality of a set $A$ is written as $n(A)$ or $|A|$. For example, if $A=\{1,2,3,4,5\}$, then $|A|=5$.

Summary of Set Types

What Are Subsets and Proper Subsets?

Often, we need to talk about sets that are contained within other sets. This is where the idea of a subset comes in.

A set $A$ is a subset of a set $B$ if every element of $A$ is also an element of $B$. We use the symbol $\subseteq$ to denote this relationship. We write $A\subseteq B$.

For example, if $A=\{1,3\}$ and $B=\{1,2,3,4\}$, then $A\subseteq B$ because both $1$ and $3$ are in set $B$. However, $B$ is not a subset of $A$ (written $B⊈A$) because $2$ and $4$ are in $B$ but not in $A$.

Two important rules for subsets:

1. Every set is a subset of itself (e.g., $A\subseteq A$).
2. The empty set $\mathrm{\varnothing }$ is a subset of every set (e.g., $\mathrm{\varnothing }\subseteq A$).

A set $A$ is a proper subset of a set $B$ if $A$ is a subset of $B$, but $A$ is not equal to $B$. This means $B$ must contain at least one element that is not in $A$. We use the symbol $\subset$ to denote a proper subset. We write $A\subset B$.

Using our example above, $\{1,3\}\subset \{1,2,3,4\}$ because it's a subset and it's not equal to the larger set. However, $\{1,2,3,4\}$ is NOT a proper subset of $\{1,2,3,4\}$.

How Do We Combine Sets? Union and Intersection

Just like we can add or multiply numbers, we can perform operations on sets. The two most common operations are union and intersection.

Union of Sets ( $\cup$ )

The union of two sets, $A$ and $B$, is a new set that contains all the elements that are in set $A$, or in set $B$, or in both. The key word here is "OR". The symbol for union is $\cup$. We read $A\cup B$ as "A union B".

When you combine the sets, remember not to list any duplicate elements.

Intersection of Sets ( $\cap$ )

The intersection of two sets, $A$ and $B$, is a new set that contains only the elements that are in BOTH set $A$ and set $B$. The key word here is "AND". The symbol for intersection is $\cap$. We read $A\cap B$ as "A intersection B".

If two sets have no elements in common, their intersection is the empty set. Such sets are called disjoint sets.

What Is the Difference of Sets and the Complement?

Besides combining sets, we can also talk about what's left over after taking elements away. This leads to the operations of difference and complement.

Difference of Sets ( $A-B$ )

The difference between two sets, written as $A-B$, is the set of all elements that are in set $A$ but are NOT in set $B$. Think of it as starting with set $A$ and removing everything that is also in $B$.

Note that $A-B$ is generally not the same as $B-A$. The order matters!

Complement of a Set ( $A^{\prime }$ )

The complement of a set $A$, denoted as $A^{\prime }$ or $A^c$, is the set of all elements in the universal set $U$ that are NOT in set $A$. To find the complement, you must know what the universal set $U$ is for that problem.

Essentially, the complement is everything *outside* of set A.

Common Mistakes to Avoid with Sets

As you get comfortable with sets, it's easy to fall into a few common traps. Keep an eye out for these potential mistakes:

• Confusing $\mathrm{\varnothing }$ and $\{\mathrm{\varnothing }\}$: The symbol $\mathrm{\varnothing }$ represents the empty set—a set with zero elements. The notation $\{\mathrm{\varnothing }\}$ represents a set that contains one element, and that element happens to be the empty set. They are not the same.
• Repeating Elements: Remember that elements in a set are distinct. Writing $\{a,b,b,c\}$ is incorrect; it should simply be $\{a,b,c\}$. The repetition is ignored.
• Mixing Up Subset Symbols: The symbol $\subseteq$ means "is a subset of" (and could be equal), while $\subset$ means "is a proper subset of" (and cannot be equal). Be precise with your notation.
• Confusing Union and Intersection: A simple way to remember is that Union starts with 'U' and its symbol $\cup$ looks like a 'U'. Intersection's symbol $\cap$ looks like an 'n', as in 'and'—it contains elements in set A and set B.
• Forgetting the Empty Set is a Subset: It's a fundamental rule that is easy to forget when listing all subsets of a set. The empty set $\mathrm{\varnothing }$ is a subset of every single set.
• Confusing Element and Subset: The symbol $\in$ is used for elements, while $\subseteq$ is for subsets. For the set $A=\{5,6\}$, it is correct to say $5\in A$ and $\{5\}\subseteq A$, but it is incorrect to say $5\subseteq A$ or $\{5\}\in A$.

Quick Reference Summary

Here is a quick summary table of the key terms and symbols covered in this lesson. Use it as a handy reference as you practice working with sets.