I have recently went through the definition of the Greatest Lower Bound (GLB) and Least Upper Bound (LUB) and I was somewhat confused with regards to the definition of the GLB in this book of mine.

For any Greatest Lower Bound of $S$, $m$:

$$glb(S)=m\iff [(\forall s\in S)(s\le l)]\to(m\ge l)$$

For any Least Upper Bound of $S$, $M$:

$$lub(S)=M\iff[(\forall s\in S)(s\le L)]\to(M\le L)$$

My question on the GLB is why $(s\le l)$ instead of $(s\ge l)$.

I mean, if $(s\le l)$, it would not seem to make sense to me as $l$ will be greater than every $s$ in the set $S$, which essentially tells me that $l$ is located in the upper bound of the set $S$.

Furthermore, the consequent $(m\ge l)$ would further mean that $m$ is located at the upper bounds of $S$, resulting in a contradiction as the definition of $m$ is the Greatest Lower Bound of $S$.

Would appreciate it lots if someone can solve this issue for me on if the definition for GLB is correct or not, thank you very much for the help!

  • $\begingroup$ Typo? Anyway, the "$m = \dotsc$" stuff is not right, $m$ is (usually) not a logical formula/proposition. $\endgroup$ – Daniel Fischer May 10 '16 at 15:58
  • $\begingroup$ Edited, apologies it should be a bi-implication instead of a equal sign, hope the changes clarifies the question, thank you. $\endgroup$ – Derp May 10 '16 at 16:05
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    $\begingroup$ There is indeed a typo in the definition: it should read $\ell\le s$ (or $s\ge\ell$). The intent is to say that if $\ell$ is a lower bound for $S$, then $m\ge\ell$. $\endgroup$ – Brian M. Scott May 10 '16 at 16:07
  • $\begingroup$ MathJax tip: \iff gives $\iff$. Also, \inf and \sup are there for infimum and supremum. $\endgroup$ – Daniel Fischer May 10 '16 at 16:07
  • $\begingroup$ Definitely a typo. I don't like the logic symbols and don't think they are right though they may be. The first is saying "m is the numbers such that for all s in S, s is less then or equal to l then it must follow m is less than or equal to m". That's okay (albeit it convoluted). The next says "M is the numbers such that for all s in S, s is less then or equal to L then it must follow L is greater than or equal to M". So M could be any point in S, any point less than any point in S, any point lesst than or equal to L. $\endgroup$ – fleablood May 10 '16 at 16:08

You're right -- there is an error in the statement. Presumably, $l$ is a lower bound for $S$, so it should be true that $l\leq s$ for each $s\in S$.

  • $\begingroup$ Alright got it, thank you all for the clarification! $\endgroup$ – Derp May 10 '16 at 16:16

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