5.1 Sets

A $set$ is an unordered collection of objects, called $elements$ or $members$ of the set. A set is said to $contain$ its elements.

$x \in A$ denotes that x is an element of the set A.
$x \notin A$ denotes that x is not an element of the set A.

\begin{aligned} N & : = {0, 1, 2, 3, \dots} The Set of Natural Numbers \\ Z & : = {0, 1, - 1, 2, - 2, 3, - 3, \dots} The Set of Integers \\ N^{+} & : = {1, 2, 3, \dots} The Set of Positive Natural Numbers \\ Q & : = {\frac{a}{b} | a, b \in Z, b \neq 0} The set of Rational Numbers \\ R & : = The set of Real Numbers \\ R^{+} & : = The set of Positive Real Numbers \end{aligned}

A : = {0, 1, 2, 3, 4, 5} = {0, 1, 2, \dots, 5}

This is called the roster method ( or extensional set notation )

\begin{array}{r} B : = {x | x \in A a n d x i s e v e n} \end{array}

This is called the set builder notation ( or intentional set notation )

The $empty set$ is the ( uniquely determined ) set that contains no element(s)
We denote it by $\emptyset$ , or by ${}$

A set with exactly one element is also called a $singleton set$

Let A, B be sets

A is a $subset$ of B (short: $A \subseteq B$ ), if every element of A is also an element of B
A is a $proper subset$ of B (short: $A ⊊ B$ ), if $A \subseteq B$ and $A \neq B$
A is a $superset$ of B (short: $A \supseteq B$ ), if $B \subseteq A$
A is a $proper superset$ of B (short: $A ⊋ B$ ), if $A \supseteq B$ and $A \neq B$

Alternative notation for $⊊ :$ $\subset$