least upper bound etc.

**robocop_911** · June 13th 2008, 10:11 AM

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.

**kalagota** · June 13th 2008, 07:04 PM

Originally Posted by robocop_911

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 $S$ is any element $r \in \mathbb{R}$ such that if you take any element $s \in S$ , then $s \leq r$

b. a lower bound of the set $S$ is any element $r \in \mathbb{R}$ such that if you take any element $s \in S$ , then $r \leq s$

note that upper bounds and lower bounds are not necessarily in $S$ .

c. the least upper bound of $S$ , or we call the supremum of S $\sup S$ is an upper bound $u$ of $S$ such that if you take any upper bound $u_0$ of $S$ , $u \leq u_0$ . In other words, $\sup S$ is the smallest among all the upper bounds.

d. the greatest lower bound of $S$ , or we call the infimum of S $\inf S$ is a lower bound $v$ of $S$ such that if you take any lower bound $v_0$ of $S$ , $v_0 \leq v$ . In other words, $\inf S$ is the biggest (greatest) among all the lower bounds.

Example:
Consider the set $\{1, 2, 3, 4, 5\}$

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: $A := \left\{ {\frac{1}{n} : n=1,2,3,...} \right\}$

you should see that 1 is the supremum of $A$ and 0 is its infimum.

**ThePerfectHacker** · June 14th 2008, 06:46 PM

Originally Posted by robocop_911

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.

Let $(A, \leq)$ be a partially ordered set. Let $X\subset A$ be a non-empty subset. We say $a\in A$ is an upper bound for $X$ if $x\leq a$ for all $x\in X$ . Similarly we say $b\in A$ is a lower bound for $X$ if $b\leq x$ for all $x\in X$ . Note we do not require that $a,b\in X$ , they do not have to. We define the least upper bound to be $a\in A$ , an upper bound, so that if $x\leq b$ for all $x\in A$ then $a\leq b$ , hence the name "upper bound". Look at what kalagota did, though we was working with $\mathbb{R}$ rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.

**robocop_911** · June 15th 2008, 10:10 AM

Originally Posted by ThePerfectHacker

Let $(A, \leq)$ be a partially ordered set. Let $X\subset A$ be a non-empty subset. We say $a\in A$ is an upper bound for $X$ if $x\leq a$ for all $x\in X$ . Similarly we say $b\in A$ is a lower bound for $X$ if $b\leq x$ for all $x\in X$ . Note we do not require that $a,b\in X$ , they do not have to. We define the least upper bound to be $a\in A$ , an upper bound, so that if $x\leq b$ for all $x\in A$ then $a\leq b$ , hence the name "upper bound". Look at what kalagota did, though we was working with $\mathbb{R}$ rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.

How do we find out if x<= a or x>=a if the set consists of {a,b,c,d...}.

Should it be lexicographic?

**kalagota** · June 15th 2008, 08:15 PM

Originally Posted by robocop_911

How do we find out if x<= a or x>=a if the set consists of {a,b,c,d...}.

Should it be lexicographic?

i think so.. most of the time, the alphabet is mapped to the set of natural numbers, i.e., 1 for a, 2 for b, etc.. but also consider what your teacher told you about the mapping..

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