Subsets and Intervals as Subsets of R

Subsets. A set $A$ is called a subset of a set $B$, written $A \subseteq B$,

if every element of $A$ is also an element of $B$. Formally,

If $A \subseteq B$ but $A \ne B$ (that is, $B$ has at least one element not in $A$), we

say $A$ is a proper subset of $B$, written $A \subset B$.

Two basic facts follow directly from the definition:

• Every set is a subset of itself: $A \subseteq A$.
• The empty set is a subset of every set: $\varnothing \subseteq A$, for any set $A$. (This holds vacuously — there is no element of $\varnothing$ that could fail to be in $A$.)

Worked illustration. If $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4, 5\}$, then every

element of $A$ (namely $1, 2, 3$) is present in $B$, so $A \subseteq B$; and since

$4 \in B$ but $4 \notin A$, in fact $A \subset B$ (a proper subset).

Subsets of the set of real numbers: intervals. Certain subsets of $\mathbb{R}$

occur constantly in mathematics — those consisting of "all real numbers between two

given numbers." These are called intervals, and they have their own compact

notation. Let $a, b \in \mathbb{R}$ with $a < b$.

• Open interval: $(a, b) = \{x \in \mathbb{R} : a < x < b\}$ — excludes both endpoints.
• Closed interval: $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$ — includes both endpoints.
• Semi-open (or semi-closed) intervals:

• Infinite (unbounded) intervals:

and $(-\infty, \infty) = \mathbb{R}$ itself. The symbols $\pm\infty$ are never

included, since they are not real numbers — a square or round bracket is never

placed next to $\infty$ or $-\infty$. …