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 $a \not \in A$. 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. 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.