3.4 Significant subsets of real numbers

In this chapter we will recall the definition of intervals and introduce some new notions describing the properties of subsets of real numbers.

Intervals represent important subsets of real numbers. For $a,b \in \mathbb{R}$, $a < b$, we define:

\begin{equation*} \begin{aligned} (a,b) &= \big\{ x \in \mathbb{R} \,\big|\, a < x < b \big\} & &\text{open interval}, \\ [ a,b ] &= \big\{ x\in\mathbb{R} \,\big|\, a \leq x \leq b \big\} & &\text{closed interval}, \\ [ a,b ) &= \big\{ x\in\mathbb{R} \,\big|\, a \leq x < b \big\} & &\text{left-closed and right-open interval}, \\ ( a,b ] &= \big\{ x\in\mathbb{R} \,\big|\, a < x \leq b \big\} & &\text{right-closed and left-open interval}, \\ ( a,+\infty) &= \big\{ x\in\mathbb{R} \,\big|\, a < x \big\} & &\text{open interval}. \end{aligned} \end{equation*}

The unbounded intervals $[ a,+\infty)$, $(-\infty, a)$ and $(-\infty, a]$ are defined analogously.

Furthermore, for subsets of the real axis, we will recall the following definition. We call a set $A\subset\mathbb{R}$ bounded from above (resp. below), if there exists a constant $K\in\mathbb{R}$ such that for every $x\in A$, $x < K$ (resp. $x > K$). We call a set $A\subset\mathbb{R}$ bounded, if it is bounded from above as well as from below.

Let $A\subset\mathbb{R}$. We call a number $a\in A$ a maximum of the set $A$, if for every $x\in A$ we have that $x \leq a$. We call a number $b \in A$ a minimum of the set $A$, if for every $x\in A$ we have that $x \geq b$. In other words, a maximum (resp. minimum) of a set $A$ of real numbers is such an element of it which is greater (resp. less) than or equal to all other elements of the set. We also denote the maximum (resp. minimum) of a set $A$ by $\max A$ (resp. $\min A$).

A maximum (respectively minimum) of a set defined in this way need not always exist. For instance, there is no minimum nor maximum of the set $(1,2)$ as numbers $1$ and $2$ do not belong to $(1,2)$. This problem can be solved by introducing an infimum and a supremum of a set which represent generalizations of minimum and maximum. We will study these notions in more detail in BIE-ZMA lectures.

Question 3.4

Which of the sets below are bounded from above, from below or bounded?

  1. $\displaystyle\Big\{\frac{1}{n} \,\Big|\, n \in \mathbb{N} \Big\}$,

  2. the set of all prime numbers,

  3. the set of all solutions of the inequation $x^2 - (\pi +1)x + \pi > 0$,

  4. $\displaystyle \{\sin x \mid x\in\mathbb{R}\}$.

Show answer

a) bounded, b) bounded from below, c) not bounded from above or below, d) bounded.

Question 3.5

Determine the maxima and minima of the following sets if they exist.

  1. $A = \{2,-1,3\}$,

  2. $B = (4,a]$, where $a > 4$ is a fixed parameter,

  3. $C = \{ (-1)^n \mid n \in \N \}$,

  4. $D = \{ 2k-3 \mid k \in \N \}$,

  5. $E = \{ 2k-3 \mid k \in \Z \}$.

Show answer

a) $\min A = -1$, $\max A = 3$, b) has no minimum, $\max B = a$, c) $\min C = -1$, $\max C = 1$, d) $\min D = -1$, has no maximum, e) has no maximum neither minimum.

Question 3.6

Prove or disprove this claim: every set which is bounded from above has a maximum.

Show answer

The claim is not true, for instance consider $(0,1)$.