Set Notation

Set Notation

Definition: A set $A$ is a collection of elements $a$. We say that $a$ belongs to $A$ or $a$ is a member of $A$ and write $a \in A$. If $a$ does not belong to $A$, we write $a \not \in A$.
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From the diagram above, we can say that $a, b, c, d, e \in A$ since the elements $a$, $b$, $c$, $d$, and $e$ are contained within the set $A$. Furthermore, we can say that $f, g \not \in A$ since these elements are not contained within the set.

Now there are two ways to define a set. The first way is by explicitly writing our all of the elements in a set. For example, consider a set $A$ that lists all of the natural numbers less or equal to 5. Therefore, $A = \{ 1, 2, 3, 4, 5 \}$. Another way to define a set is by defining the set with a property $P(x)$, that is, $A = \{ x : P(x) \}$, which reads, "$A$ is the set containing elements $x$ such that $P(x)$ is true." In our example, we could write $A = \{ x : x \in \mathbf{N} \: \mathrm{and} \: x ≤ 5 \}$.

We also have some common symbols that denote special types of numbers:

$\mathbf{R}$ The set of real numbers.
$\mathbf{Z}$ The set of integers, $\mathbf{Z} = \{..., -2, -1, 0, 1, 2, ... \}$.
$\mathbf{Z}^+$ The set of positive integers, $\mathbf{Z}^+ = \{ 1, 2, 3, ... \}$.
$\mathbf{N}$ The set of natural numbers, $\mathbf{N} = \{ 1, 2, 3, ... \}$.
$\mathbf{Q}$ The set of rational numbers, $\mathbf{Q} = \{ \frac{a}{b} : a, b \in \mathbf{Z} \: and \: b ≠ 0 \}$.
$\mathbf{C}$ The set of complex numbers $\mathbf{C} = \{ a + bi : a, b \in \mathbf{R} \}$.

Subsets and Supersets of a Set

Definition: A set $B$ is said to be a subset of $A$ written $B \subseteq A$ if all elements of $B$ are members of $A$, that is, if $b \in B$ then $b \in A$. Alternatively, we can say that $A$ is a superset of $B$ written $A \supseteq B$ to mean the same thing.
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For example, suppose that $A = \{ 1, 3, 5, 7, 9 \}$ and $B = \{ 1, 3, 5 \}$. Since all elements of $B$ are also in $A$, we can say that $B \subseteq A$. In our diagram above, we note that $A = \{ a, b, c, d, e \}$ and $B = \{ b, d \}$ so $B \subseteq A$.

We should also note two other important definitions regarding sets very similar to the last definitions.

Definition: A set $B$ is said to be a proper subset of $A$ written $B \subset A$ if all elements of $B$ are members of $A$ but $B \neq A$. Similarly, we can say that $A$ is a proper superset of $B$ written $A \supset B$ to mean the same thing.

For example, consider the sets $A = \{ 1, 2, 3 \}$, $B = \{ 1, 2, 3 \}$ and $C = \{ 1, 2 \}$. We can say that $[[$ B$ and $C$ are subsets of $A$, that is $B \subseteq A$ and $C \subseteq A$. However, we cannot say that $B$ is a proper subset of $A$ since $A = B$. We can say that $C$ is a proper subset of $A$ though, that is $C \subset A$.

Equality of Sets

Definition: Two sets $A$ and $B$ are said to be equal if and only if $A \subseteq B$ and $B \subseteq A$. If $A$ and $B$ are equal, then we write $A = B$.

For example, consider the set of natural numbers $\mathbb{N} = \{ 1, 2, ... \}$ and the set of positive integers $\mathbb{Z}^+ = \{ 1, 2, ... \}$. We note that $\mathbb{N} \subseteq \mathbb{R}^+$ and $\mathbb{R}^+ \subseteq \mathbb{N}$. In other words, the set of natural numbers is fully contained in the set of positive integers and the set of positive integers is fully contained in the set of natural numbers. Therefore we can write that these sets are equal, that is $\mathbb{N} = \mathbb{R}^+$.

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