Set Notation and Operations

What is a Set?

At its core, a set is a well-defined collection of distinct objects, which we call elements or members. The key characteristics of a mathematical set are:

1. Unordered: The order in which elements are listed does not matter. The set $\{1,2,3\}$ is identical to the set $\{3,1,2\}$.
2. Distinct Elements: A set does not contain duplicate elements. Even if an element is listed multiple times, it is considered to be in the set only once. The set $\{a,a,b,c,c\}$ is identical to $\{a,b,c\}$.

To get a quick overview of these foundational ideas, please watch the first few minutes of the following video.

An element is either in a set or it is not. We use the symbol $\in$ to denote that an element belongs to a set, and $\notin$ to denote that it does not. For example, if $A=\{1,2,3\}$, then $2\in A$ but $4\notin A$.

Representing Sets: From Lists to Rules

To work with sets, we need precise ways to describe them. There are a few standard methods, each useful in different contexts.

A comprehensive resource for this topic is the text "Sets and Set Theory" from the University of Western Ontario's Discrete Structures course. We will refer to it multiple times.

To summarize the key methods:

• Roster Method: Explicitly listing all elements, e.g., $A=\{\text{red, yellow, blue}\}$.
• Set-Builder Notation: Defining a set by a rule, e.g., $E=\{x\in \mathbb{Z}\mid x\text{ is even}\}$, which reads "the set of all elements $x$ in the integers $\mathbb{Z}$ such that $x$ is even."
• Interval Notation: A compact way to denote sets of real numbers, e.g., $[0,1)$ represents the set $\{x\in \mathbb{R}\mid 0\le x<1\}$.

Fundamental Set Properties and Relationships

Now that we know how to represent sets, let's explore their fundamental properties and the relationships between them.

Let's consolidate these important concepts:

• Equality ($A=B$): Two sets are equal if and only if they contain the exact same elements. As we'll see, this is formally proven by showing that $A\subseteq B$ and $B\subseteq A$.
• Subset ($A\subseteq B$): Set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$. If $A$ is a subset of $B$ but not equal to $B$, it is a proper subset ($A\subset B$).
• Empty Set ($\varnothing$): The unique set that contains no elements. By definition, the empty set is a subset of every set. Its cardinality is 0: $|\varnothing |=0$.
• Universal Set ($U$): The set of all possible elements under consideration in a given context.
• Cardinality ($|A|$): The number of distinct elements in a finite set. For instance, if $A=\{1,2,3\}$, then $|A|=3$.
• Power Set ($\mathcal{P}(A)$): The set of all possible subsets of $A$. If $|A|=n$, then $|\mathcal{P}(A)|=2^n$. For example, if $A=\{1,2\}$, its subsets are $\varnothing ,\{1\},\{2\},\{1,2\}$. Therefore, $\mathcal{P}(A)=\{\varnothing ,\{1\},\{2\},\{1,2\}\}$.

Operations on Sets

Much like we perform arithmetic operations on numbers, we can perform operations on sets to create new sets. Venn diagrams are an excellent way to visualize these operations.

The four fundamental operations are union, intersection, difference, and complement.

Here is a formal summary of these operations:

1. Union ($A\cup B$): The set of all elements that are in set $A$, or in set $B$, or in both.

   $$A\cup B=\{x\mid x\in A\vee x\in B\}$$

   Example: If $A=\{1,2\}$ and $B=\{2,3\}$, then $A\cup B=\{1,2,3\}$.

2. Intersection ($A\cap B$): The set of all elements that are in both set $A$ and set $B$.

   $$A\cap B=\{x\mid x\in A\wedge x\in B\}$$

   Example: If $A=\{1,2\}$ and $B=\{2,3\}$, then $A\cap B=\{2\}$. If the intersection is the empty set ($A\cap B=\varnothing$), the sets are called disjoint.

3. Difference ($A\setminus B$): The set of all elements that are in set $A$ but not in set $B$.

   $$A\setminus B=\{x\mid x\in A\wedge x\notin B\}$$

   Example: If $A=\{1,2\}$ and $B=\{2,3\}$, then $A\setminus B=\{1\}$.

4. Complement ($A^c$ or $\overline{A}$): The set of all elements in the universal set $U$ that are not in set $A$.

   $$A^c=\{x\in U\mid x\notin A\}=U\setminus A$$

   Example: If $U=\{1,2,3,4,5\}$ and $A=\{1,2\}$, then $A^c=\{3,4,5\}$.

Set Identities and Proofs

As you become more familiar with these operations, you will encounter set identities, which are statements of equality that are always true. For example, the distributive laws and De Morgan's laws are fundamental.

• Distributive Law: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
• De Morgan's Law: $(A\cup B{)}^c=A^c\cap B^c$

Because you are interested in building a rigorous foundation, it's valuable to see how such identities are proven. The standard method is to show that the set on the left is a subset of the set on the right, and vice-versa, which establishes their equality.

The video you just referred to contains an excellent, step-by-step proof of a distributive law. Watching this will connect the abstract definitions to the formal practice of mathematical proof.

Conclusion

In this lesson, we have built a solid foundation in the language of set theory. You've learned how to define and represent sets using roster, set-builder, and interval notations. We explored fundamental properties like cardinality and subsets, and you now know how to manipulate sets using the core operations of union, intersection, difference, and complement.

Key Takeaways:

• A set is an unordered collection of unique elements.
• Sets can be represented by listing elements (roster method) or by defining a property (set-builder notation).
• Key concepts include subsets ($\subseteq$), cardinality ($|\cdot |$), the empty set ($\varnothing$), and the power set ($\mathcal{P}(A)$).
• The main set operations are union ($\cup$), intersection ($\cap$), difference ($\setminus$), and complement ($A^c$).
• Set identities, like the distributive and De Morgan's laws, can be proven formally by showing mutual subset relationships.

In our next lesson, we will shift gears to another essential topic: logarithms and exponentials. The concepts from today will reappear when we formally define the domain and range of these functions, reinforcing the idea that set theory is the bedrock upon which other mathematical structures are built.