Supremum of Subset of Real Numbers/Examples
Examples of Suprema of Subsets of Real Numbers
Example 1
The subset $S$ of the real numbers $\R$ defined as:
- $S = \set {1, 2, 3}$
admits a supremum:
- $\sup S = 3$
Example 2
The subset $T$ of the real numbers $\R$ defined as:
- $T = \set {x \in \R: 1 \le x \le 2}$
admits a supremum:
- $\sup T = 2$
Example 3
The subset $V$ of the real numbers $\R$ defined as:
- $V := \set {x \in \R: x > 0}$
does not admit a supremum.
Example 4
Consider the set $A$ defined as:
- $A = \set {3, 4}$
Then the supremum of $A$ is $4$.
However, $A$ contains no element $x$ such that:
- $3 < x < 4$.
Example: $\openint \gets 0$
Let $\R_{<0}$ be the (strictly) negative real numbers:
- $\R_{<0} := \openint \gets 0$
Then the supremum of $\R_{<0}$ is $0$.
Example: $\hointl 0 1$
Let $\hointl 0 1$ denote the left half-open real interval:
- $\hointl 0 1 := \set {x \in \R: 0 < x \le 1}$
Then the supremum of $\R_{<0}$ is $1$.
Example 5
The subset $S$ of the real numbers $\R$ defined as:
- $S = \set {x \in \R: x^2 \le 2 x - 1}$
admits a supremum:
- $\sup S = 1$
such that $\sup S \in S$.
Example 6
The subset $S$ of the real numbers $\R$ defined as:
- $S = \set {x \in \R: x^2 + 2 x \le 1}$
admits a supremum:
- $\sup S = -1 + \sqrt 2$
such that $\sup S \in S$.
Example 7
The subset $S$ of the real numbers $\R$ defined as:
- $S = \set {x \in \R: x^3 < 8}$
admits a supremum:
- $\sup S = 2$
such that $\sup S \notin S$.
Example 8
The subset $S$ of the real numbers $\R$ defined as:
- $S = \set {x \in \R: x \sin x < 1}$
does not admit a supremum.