Interactive · 3 questions
Some sets of numbers have "bounds". That just means that the entire set is on one side of that number.
If every number in the set is less than or equal to the bound, the bound is an upper bound. If every number in the set is greater than or equal to the bound, the bound is a lower bound.
If a set of numbers has an upper bound, we say it is bounded from above; if a set of numbers has a lower bound, we say it is bounded from below.
Related concepts are the maximum and the minimum of a set, which are just the largest and smallest elements, respectively. Note that the maximum of a set is always also an upper bound of the set, and the minimum of a set is always also a lower bound of the set.
Which of the following are upper bounds of the set ?
Which of the following is the maximum of the set ?
In fact, every finite set of numbers has a maximum and a minimum... except one. <* maybe prove this with induction someday *> Can you think of what it is?
The empty set does not have a maximum or minimum, because it doesn't have any members that might qualify. However, it does have lots of upper and lower bounds.
Which of the following are upper bounds of the empty set?
Every number is an upper bound of the empty set, as well as a lower bound. This is because "every number in the empty set" vacuously satisfies any inequality we'd care to name, since there aren't any.
These ideas are all very simple when we only deal with finite sets, but things get more interesting with infinite sets.