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What is the definition of bounds?

I read somewhere that in the set: $\{3,5,11,20,22\}$, $3$ is a lower bound, and $22$ is an upper bound. In the same explanation it is said that $2$ is also a lower bound, but it does not mention wether $45$ is also an upper bound. Is $45$ an upper bound as well? (Im assuming the author wrote it for numbers $\in\mathbb{N}$. Does this last bit matter?

In the same explanation there is an example of an human, about how tall that human is. It's noted that it can't be lesser than $0$, so $0$ is a lower bound. Suppose that this human can be negative in length, i.e. it's length is some integer defined as $x\in\mathbb{Z}$. If we have a set of human lengths, though the number of humans can vary: $\{-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7\}$ Can the lower bound be $-7$ feet tall? Would it mean that the lengths also vary?

  • In the first set anything lower or equal than $3$ is a lower bound and anything bigger or equal than $22$ is a upper bound. – kingW3 Aug 29 '21 at 01:12
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    What makes $3$ and $22$ special here is that $3$ is the greatest lower bound, and $22$ is the least upper bound. – user2661923 Aug 29 '21 at 01:30

1 Answers1

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The definition of lower and upper bound might vary depending on the context, but in analysis the definition of "upper bound" is as follows:

Let $S$ be an ordered set*, and let $E$ be a subset of $S$. We say that $\alpha$ is an upper bound of $E$ if $x\le\alpha$ for all $x\in E$.

Lower bounds are defined in the same way. So in your first example, we can take $S$ to be the set of positive integers, and $E=\{3,5,11,20,22\}$. Then, $42$ is an upper bound of $E$, because $x\le42$ for all $x\in E$.

*An ordered set is just a collection of elements with a relation "$<$" defined on them. For example, $\mathbf R$ is an ordered set if we define $a<b$ to mean "$b-a$ is a positive real number".

Joe
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