Introduction to Basic Set Notation

The Membership Symbol ($\in$)

To say that an object is an element of a set, we use the symbol $\in$, which is read as "is an element of" or "is in".

If we use our previous example of the vowel set $V = {a, e, i, o, u}$, we can make the following mathematical statements:

• $a \in V$ (read as "$a$ is an element of $V$")
• $e \in V$ (read as "$e$ is in $V$")

The Non-Membership Symbol ($\notin$)

Conversely, if an object does not belong to a set, we use the symbol $\notin$, which is read as "is not an element of" or "is not in".

Using the same set $V$, we can state:

• $b \notin V$ (read as "$b$ is not an element of $V$")
• $z \notin V$ (read as "$z$ is not in $V$")

Exam Tip: When checking for membership, always look closely at the definition of the set. A common mistake is to assume an element belongs to a set because it "seems" like it should, rather than verifying it against the specific elements listed or the rule provided.

Worked Example: Applying Notation

Let's put these concepts together with a practical example. Suppose we define a set $S$ consisting of the first five positive even integers.

First, we list the elements to define the set:

$$S = \{2, 4, 6, 8, 10\}$$

Now, let's test several statements for correctness using our notation:

1. Is $4 \in S$? Yes, because $4$ is one of the numbers listed in the set. Statement: $4 \in S$ (True)

2. Is $5 \in S$? No, $5$ is an odd number and is not in our list. Statement: $5 \notin S$ (True)

3. Is $10 \in S$? Yes, $10$ is the final element in our set. Statement: $10 \in S$ (True)

4. Is $0 \in S$? No, we defined the set as containing positive even integers, and $0$ is not positive. Statement: $0 \notin S$ (True)

Summary Table

To help you study, here is a quick reference for the basic notation we have covered:

By mastering these symbols, you have built the foundation necessary to discuss more advanced topics like subsets, unions, and intersections in set theory.