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Ans. C

in Mathematical Logic by Loyal (7k points) | 128 views
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Please Explain.
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A) false as its poset but lattice too(every pair of element will have a max and min)

b)false its  distributive lattice as distributive lattice has atmost 1 complement.

c)true as its distributive and n>0 (givenn  positive integer ) and its countably infinite so not bounded.

d) false : its a poset as it satisfies reflexive,antisymm,transitive property.
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@arvin

No element has even one complement right?

 It's hasse diagram will be chain right then it will be TOS?

Can you explain how it is reflexive?
+2

yes (distributive lattice says every element should have atmost 1 complement)

yes ( it will always be a chain)

its reflexive because every element will be the divisor of itself...

for eg : (1|1) (2|2) (4|4) .........................-->aRa =1 always  

+1
Thanks, in this example is there any element with one complement.?

I thinks there is not
+1
@rishav no bro no element has any complement.
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how it will be distributive?
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because it has no complement or 0 complement... and distributive lattice has atmost 1complement.
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Addition to Arvin, it is TOS in hasse diagram it will be a chain so, no presence of L1* and L2*
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is it a poset too?
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Yes mam
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how??
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Because it is partial ordering relation, because it is Reflexive, antisymmetric and transitive.
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say for (2,4) how it will be antisymmetric?
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Because we can't find any pair (4,2)
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2R4=2 which is one among the domain.

But 4R2=.5 which is not in the domain.
So it's antisymmetric.
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4R2 not 4??
+1
no mam here in the question they have mentioned that n is a +ve integer. and a is divisor of b. but 4 is not divisor of 2.
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yes, it is a chain

every 2 element has a LUB and a GLB, that is why it is lattice

but it no lower bound bound and no upper bound

So, it is not bounded

that means the lattice has  LUB but no upper bound

And can u tell me is it complemented lattice or not?

I think it is complemented lattice too
+2
its not a complemented lattice as to be complemented it has to satisfy two properties:
1)it should be bounded
2)it should have atleast one complement for each element.
but it fails on both the criteria so its not a complemented lattice.
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Statement should be "it should have atleast one complement for every element"
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yes :p updated
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@arvin

distributive lattice says every element should have atmost 1 complement.

No. Only its converse is true. 

See this: https://gateoverflow.in/265533/lattice

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