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I am confused about upper bounds of a set.

Consider a set:

$A = ${$1, 2, 3, 4, 5, 6, 7$}

How many upperbounds are there?

Does the upperbound need to be in the set?

Also about supremum. What is the difference between a supremum and an upper bound?

Andrés E. Caicedo
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Anon
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  • An upperbound is any number greater than or equal to every number in A. Here A has a greatest element, 7, so any number $ge$ 7 will be an upperbound for A and, except for 7, it will not be in A. The supremum is the least upperbound. It may or may not be it A. If it is in A, it's the greatest element of A. – Bernard Dec 30 '14 at 15:30

1 Answers1

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An upper bound is any number $U$ with the property that $a\leq U$ for each element $a$ of $A$.

For example, $10$ is an upper bound for your $A$ because $1\leq 10$, $2\leq 10$, $3\leq 10$, $4\leq 10$, $5\leq 10$, $6\leq 10$, and $7\leq 10$.

Obviously, if you find one number $U$ that is an upper bound for $A$, then also every number greater than $U$ is also an upper bound. That's because if $V>U$, then $a\leq U$ implies $a\leq V$.

But, among all of the upper bounds, there will be a smallest upper bound (assuming the set $A$ is not empty). That number is called the least upper bound of $A$. The term supremum is just another word for least upper bound. You will see this written variously as $$\sup A$$ or $$\operatorname{lub} A$$

In your case, the least upper bound is $7$. Any time a set has a maximum, this maximum coincides with the least upper bound. For example, every finite nonempty set has a maximum.

But if the set is infinite, the maximum value might happen to be missing from the set. For example, the set of numbers $$\{0.9, 0.99, 0.999, 0.9999,\cdots\}$$ contains elements that get arbitrarily close to $1$, but $1$ itself is not in the set. $1$ is an upper bound for the set, and it is the least upper bound for the set, but it is not the maximum for the set. Technically, the maximum of a set must belong to that set.

MPW
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