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 $\displaystyle S$ is any element $\displaystyle r \in \mathbb{R}$ such that if you take any element $\displaystyle s \in S$, then $\displaystyle s \leq r$
b. a lower bound of the set $\displaystyle S$ is any element $\displaystyle r \in \mathbb{R}$ such that if you take any element $\displaystyle s \in S$, then $\displaystyle r \leq s$
note that upper bounds and lower bounds are not necessarily in $\displaystyle S$.
c. the least upper bound of $\displaystyle S$, or we call the supremum of S $\displaystyle \sup S$ is an upper bound $\displaystyle u$ of $\displaystyle S$ such that if you take any upper bound $\displaystyle u_0$ of $\displaystyle S$, $\displaystyle u \leq u_0$. In other words, $\displaystyle \sup S$ is the smallest among all the upper bounds.
d. the greatest lower bound of $\displaystyle S$, or we call the infimum of S $\displaystyle \inf S$ is a lower bound $\displaystyle v$ of $\displaystyle S$ such that if you take any lower bound $\displaystyle v_0$ of $\displaystyle S$, $\displaystyle v_0 \leq v$. In other words, $\displaystyle \inf S$ is the biggest (greatest) among all the lower bounds.
Example:
Consider the set $\displaystyle \{1, 2, 3, 4, 5\}$
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: $\displaystyle A := \left\{ {\frac{1}{n} : n=1,2,3,...} \right\}$
you should see that 1 is the supremum of $\displaystyle A$ and 0 is its infimum.
Let $\displaystyle (A, \leq)$ be a partially ordered set. Let $\displaystyle X\subset A$ be a non-empty subset. We say $\displaystyle a\in A$ is an upper bound for $\displaystyle X$ if $\displaystyle x\leq a$ for all $\displaystyle x\in X$. Similarly we say $\displaystyle b\in A$ is a lower bound for $\displaystyle X$ if $\displaystyle b\leq x$ for all $\displaystyle x\in X$. Note we do not require that $\displaystyle a,b\in X$, they do not have to. We define the least upper bound to be $\displaystyle a\in A$, an upper bound, so that if $\displaystyle x\leq b$ for all $\displaystyle x\in A$ then $\displaystyle a\leq b$, hence the name "upper bound". Look at what kalagota did, though we was working with $\displaystyle \mathbb{R}$ rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.