# What does bounded above or below mean in precaculus?

Apr 25, 2017

See explanation.

#### Explanation:

Definitions:

A set is bounded above by the number $A$ if the number $A$ is higher than or equal to all elements of the set.

A set is bounded below by the number $B$ if the number $B$ is lower than or equal to all elements of the set.

Examples:

Example 1

A set of natural numbers $\mathbb{N}$ is bounded below by the number $0$ or any negative number because for all natural numbers $n$ we have: $0 \le n$ and for every negative number $N$ we have $N \le n$

Example 2
Let $A$ be a set $A = \left\{\frac{1}{n} : n \in \mathbb{N}\right\}$

This set can be written as $A = \left\{1 , \frac{1}{2} , \frac{1}{3} , \ldots\right\}$

This set is bounded above by $1$ (or any number greater than $1$) and bounded below by $0$ (or any number lower than $0$). For all $x \in A$ we can write that: $0 \le x \le 1$

Apr 25, 2017

suppose you have a set S .

#### Explanation:

Suppose you have a set of values containing values between a real number a and b including a and b .
$S = \left\{x : x \in \left[a , b\right]\right\}$ and a is less than b .
each value in above mentioned set of values is more than or equal to a and similarly, each value is less than or equal to b.
Thus the set is said to be bounded , where ,
a is said to be the lower bound of set and b the upper bound of set .
if the set is : $S = \left\{x : x \in \left(a , b\right)\right\}$ and a is less than b .
still a is the lower bound and b the upper bound .