complete lattice and bounded lattice

Question

Prove that every complete lattice is bounded lattice but not vice-versa .

Answer 1

okay I explain you in my way !!

Complete lattice means partially ordered set in which all subsets have both a join and an meet

Bounded Lattice means  In a lattice if upper bound and lower bound exists then it's called bounded lattice

every complete lattice is Bounded lattice is easy to analyze by the definition of Complete lattice 

But every bounded lattice is not complete lattice  is the real deal here

M = {a $\varepsilon$ Q }  where  - $\sqrt{3}$ < a <  + $\sqrt{3}$  Q---- > rational number

it's bounded right ???

Now in this Hasse Diagram can you tell me that what it's  element at the Top position (dot at the top ) which is less than    $\sqrt{3}$ and what it's GLB   ?? 

you can't tell that right ??? because there's  a infinite numbers 

Though you may have a set which have Greatest and lower bound but its doesn't mean that the subset will have  these lower and upper bound

 

remember : whenever there's a infinite sets ---- > it's not a complete lattice                                             

 

Comments

@ thanks magma

lets suppose i have taken a subset { 1/3, 0 ,-1/2 }

it have upper bound 1/3 and lower bound -1/2

similarly i can take any subset and find the upper bound and lower bound.

it means i did not understand this statement  which is written by you

( Though you may have a set which have Greatest and lower bound but its doesn't mean that the subset will have  these lower and upper bound )

can you explain this statement a bit more

---

lets suppose i have taken a subset { 1/3, 0 ,-1/2 }  it have upper bound 1/3 and lower bound -1/2

you cannot take it like that

here subsets is on the relational number

lower bound can be -0.5 (1/2) or 0.51 or 0.52 or 0.52 ........ 0.555 or 0.56........infinity

you can't define particular sets here