Zeal Test Series 2019: Set Theory & Algebra - Lattice

Question

I am getting 3 minimal please check it 

Comments

It is 3 i also think so.

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I think 'm'  and 'n' is only minimal element

I know you consider  "j" also

I have a slightly doubt in it !

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it can be only 1 minimal element also which is "m"

someone clear my doubt in it !

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@ Magma ,no element is related to some element s ,then s is minimal element .as we can see in diagram element i is not related to anyone else .same thing in m,j also .so total 3 minimal are there.

OR we can see like this ,in Hasse diagram all edges has upward direction ,if some element has only out degree that element is minimal element because no element is related to that element  

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@prateek bhai i also did it in same way by your first definition but i am not getting this -

in Hasse diagram all edges has upward direction ,if some element has only out degree that element is minimal element because no element is related to that element  ,

how to see that which element has outdegree please clear this point

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As you can see except I,j, m all other have in degree also ,only these have out degree only

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thanks you  Prateek

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Put a ball at top most position(say b)  and let it roll downwards ....see at what  are the possible positions at which it  stops (here they could be i,j,m only)....they are the minimal elements.

For finding the maximal elements rotate the given Hasse diagram 180 degree and repeat the process.

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thanks PRATEEK BHAI

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@Satbir 

how would you check with same method for 1.maximum and minimum

2.GLB and LUB

PLEASE suggest 

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@Mayankprakash

find the minimal i.e. i,j,m then see the diagram again which element is at the lowest level among i,j and m ?

its m ,...so m is the minimum element.

rotate the daigram by 180 and repeat the above step to get the maximum element.

NOTE :- if there are 2 or more elemenet at the lowest level then neither of them will be maximum or minimum element because there can be only 1 maximum and 1 minimum element in a poset.

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GLB(meet) and LUB(join) are done wrt to a subset of the poset.

Suppose we take the set {g,h} and then calculate the Join and meet for it.

Put a ball at g and another at h and let them roll downwards. what are the points at which they meet ?

j and m

where they will meet early at j or at m ?  j right ? so j will be meet.

Flip the daigram and do it for getting join

we will get a,d,e,b. but where are they meeting early ?

at d or at e so there is not a join possible

NOTE:- there is only 1 meet and 1 join for a given subset of the poset.

Answer 1

3 is right.