Can anyone give me a good explanation of...
a) upper bound
b) lower bound
c) least upper bound
d) greatest lower bound
It's a chapter of Relations - "Partial ordering". My text doesn't give a good explanation about it.
a. an upper bound of the set is any element such that if you take any element , then
b. a lower bound of the set is any element such that if you take any element , then
note that upper bounds and lower bounds are not necessarily in .
c. the least upper bound of , or we call the supremum of S is an upper bound of such that if you take any upper bound of , . In other words, is the smallest among all the upper bounds.
d. the greatest lower bound of , or we call the infimum of S is a lower bound of such that if you take any lower bound of , . In other words, is the biggest (greatest) among all the lower bounds.
Example:
Consider the set
these are some upper bounds.. 5, 5.1, 6, 7, 6.5,.. in fact, any number greater than or equal to 5 is an upper bound. While the supremum of the set if 5 since if you take all the upper bounds, 5 is the smallest.. Hope you can tell what the lower bounds are, and what the infimum is.
another example:
you should see that 1 is the supremum of and 0 is its infimum.
Let be a partially ordered set. Let be a non-empty subset. We say is an upper bound for if for all . Similarly we say is a lower bound for if for all . Note we do not require that , they do not have to. We define the least upper bound to be , an upper bound, so that if for all then , hence the name "upper bound". Look at what kalagota did, though we was working with rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.