Consider the set of vowels in the English alphabet:

$$A=\{\text{a},\text{e},\text{i},\text{o},\text{u}\}$$

This set clearly lists all the vowels, and it is easy to determine whether a given letter is a vowel or not.

For a set to be meaningful, it must be well-defined. This means it must be clear whether any given object is an element of the set or not. For example, the set of vowels in the word "radio" is well-defined and can be written as:

$$A=\{\text{a},\text{i},\text{o}\}$$

Similarly, the "set of all days last year with temperatures below $0^{\circ }$C" is well-defined because it is based on objective, measurable data. However, the "set of all cold days last year" is not well-defined because the term "cold" is subjective and can vary from person to person.

The set of vowels in the English alphabet is:

$$A=\{\text{a},\text{e},\text{i},\text{o},\text{u}\}$$

The same set may also be written with a repeated entry:

$$A=\{\text{a},\text{a},\text{e},\text{i},\text{o},\text{u}\}$$

This notation is valid, but the repetition is redundant. Both rosters describe exactly the same set because each distinct vowel occurs in both. It is therefore clearer to list each element only once.

Consider two sets containing the vowels in the English alphabet:

$$A=\{\text{a},\text{e},\text{i},\text{o},\text{u}\}\text{ and }B=\{\text{u},\text{o},\text{i},\text{e},\text{a}\}$$

These two sets are identical because they contain the same elements, regardless of the order in which the elements are listed. Thus:

$$A=B$$

This example illustrates the concept of order independence in sets, where the arrangement of elements does not define the uniqueness of a set.

Here are a few sets described only in words, before translating them into symbols:

• "The set of vowels in the English alphabet."
• "The set of non-negative integers."
• "The set of non-negative integers strictly smaller than 6."

The same ideas can be written in roster form by listing their elements explicitly:

• The set of vowels in the English alphabet: $$\{\text{a},\text{e},\text{i},\text{o},\text{u}\}$$

• The set of non-negative integers: $$\{0,1,2,3,4,\dots \}$$

• The set of non-negative integers strictly smaller than 6: $$\{0,1,2,3,4,5\}$$

Note: The ellipsis ($\dots$) indicates that the pattern continues indefinitely.

The same examples can also be written by stating the rule an element must satisfy:

• The set of vowels in the English alphabet: $$A=\{x\mid x\text{ is a vowel in the English alphabet}\}$$

• The set of non-negative integers: $$B=\{x\mid x\text{ is a non-negative integer}\}$$

• The set of non-negative integers strictly smaller than 6: $$C=\{x\mid x\text{ is a non-negative integer and }x<6\}$$

Here are common intervals written both in interval notation and set-builder notation:

• Real numbers strictly between $0$ and $1$: $$(0,1)=\{x\in \mathbb{R}\mid 0<x<1\}$$

• Real numbers between $2$ and $5$, including both endpoints: $$[2,5]=\{x\in \mathbb{R}\mid 2\le x\le 5\}$$

• Real numbers greater than $3$: $$(3,\infty )=\{x\in \mathbb{R}\mid x>3\}$$

• Real numbers less than or equal to $0$: $$(-\infty ,0]=\{x\in \mathbb{R}\mid x\le 0\}$$

The following examples illustrate how the symbols $\in$ (is an element of) and $\notin$ (is not an element of) describe whether a value belongs to a particular set.

• The element $3$ belongs to the set because it appears among its members: $$3\in \{1,2,3,4,5\}$$

• The element $6$ does not belong to the set because it is not included among its elements: $$6\notin \{1,2,3,4,5\}$$

• The number $\pi$ is a real number, so it belongs to the set of all real numbers: $$\pi \in \mathbb{R}$$

• In this case, the elements of the outer set are themselves sets, so $\{1\}$ is one of its members: $$\{1\}\in \{\{1\},\{2\},\{3\}\}$$

• The number $1$ alone is not a member, because the set only contains sets as elements: $$1\notin \{\{1\},\{2\},\{3\}\}$$

• The fraction $\frac{7}{4}$ (equal to $1.75$) is in the interval because $1\le \frac{7}{4}<2$: $$\frac{7}{4}\in [1,2)$$

• The number $-5$ is not in this interval because it is not positive: $$-5\notin (0,\infty )$$

The following examples show how cardinality records the size of finite, countably infinite, uncountably infinite, and empty sets.

• If $A=\{\text{a},\text{e},\text{i},\text{o},\text{u}\}$, then $|A|=5$. This means that $A$ is a finite set and has five distinct elements.
• If $B=\{1,2,3,4,5,6\}$, then $|B|=6$. This means that $B$ is a finite set and has six distinct elements.
• If $C=\{1,2,3,\dots \}$, then $|C|={\aleph }_0$ (aleph-null, the cardinality of any countably infinite set). This means that $C$ is countably infinite, i.e., its elements can be listed one by one in an endless sequence (first 1, then 2, then 3, and so on).
• If $D=\mathbb{R}$, then $|D|=\mathfrak{c}$ (the cardinality of the continuum). This means that $D$ is uncountably infinite: its elements cannot be listed one by one in a sequence.
• If $E=\varnothing$, then $|E|=0$. The empty set contains no elements, so its cardinality is zero.

The following examples compare sets by checking whether every element of one set appears in the other.

1. Let $S=\{3,5,8\}$ and $T=\{5,3,8\}$. Since both sets contain the same elements: $$S\subseteq T\text{ and }T\subseteq S$$ Therefore, $S$ and $T$ are equal sets, and each is a subset of the other.

2. Let $S=\varnothing$ and $T=\{5,3,8\}$. The empty set contains no elements, so it is a subset of every set: $$S\subseteq T$$

3. Let $S=\{\text{red},\text{blue}\}$ and $T=\{\text{red},\text{blue},\text{green}\}$. Every element of $S$ is in $T$, so: $$S\subseteq T$$ Since $S\ne T$, $S$ is also a proper subset, although it can first be identified as a subset.

The following examples emphasize the extra requirement for a proper subset: the larger set must contain at least one additional element.

1. Let $S=\{\text{red},\text{blue}\}$ and $T=\{\text{red},\text{blue},\text{green}\}$. Every element of $S$ is in $T$, but $T$ has one additional element. Therefore: $$S\subset T$$ That is, $S$ is a proper subset of $T$.

2. Similarly, let $S=\{3,5,8\}$ and $T=\{5,8,3,2,6\}$. All elements of $S$ are contained in $T$, but $T$ has additional elements ($2$ and $6$). Hence: $$S\subset T$$ That is, again, $S$ is a proper subset of $T$.

3. The natural numbers form a proper subset of the integers: $$\mathbb{N}\subset \mathbb{Z}.$$ Both sets are countably infinite, so a proper subset need not have smaller cardinality when the sets are infinite.

These examples show that every element of a proper subset belongs to the larger set, but the larger set has at least one element that is not in the proper subset. A proper subset is strictly smaller by inclusion, although it need not have a smaller cardinality when the sets are infinite.