DM-Groups and Lattice-Q12

+1 vote

Let (L,\leq) be a lattice. Then for all x,y,z\in L which of the following is FALSE?

(A).x\lor y = y\lor x

(B).x\lor (x\land y) = x

(C).x\lor (y\land z) = (x\lor y)\land(x\lor z)

(D).x\land (x\lor y) = x

asked Jul 8, 2019 in Discrete Maths by gbeditor (32,710 points)
reshown Jul 9, 2019 by gbeditor

4 Answers

+3 votes
Best answer

Here \leq represents any relation, Not less than or eqal to specifically. Lattice might not satisfy distributive property so Option C is false. 

answered Jul 11, 2019 by getgatebook (34,870 points)
0 votes
{L,<=} is tos and all tos is always distribute lattic.

And all lattice follow Identity property , associative,commutative, absorption

But not distributed property if a lattice follow distributed property than this type of lattice called distributed lattice.

Here given lattice is tos and tos always follow all these propertie so I think all option are wrong.

Please correct me if my statement is wrong.
answered Jul 9, 2019 by (34,820 points)
why a lattice will follow C?
you are absolutely correct. i too have same dilemma. i think all options are correct
Not all lattice follow c but given lattice follow all propertie
I think it is not a description but just a notation. Every lattice may or may not follow the distributed property. I think option C is right.
Yes exactly

That less than equal to sign denotes comparability between two elements in a poset
0 votes
I think C is wrong , C holds true for distributive lattice
answered Aug 12, 2019 by (400 points)
Yes, but all other options holds true for all kind of lattices.
Only option C needs the special kind of lattice i.e Distributed lattices
0 votes
Option A is true, as it is just shuffling x and y

For Option B and D, we can have 3 kinds of cases.

Case 1: xRY, i.e x is below y in the lattice.

Case 2: yRx, i.e y is below x in the lattice.

Case 3: x and y are not related, but x and y have GLB and LUB as the structure is a lattice.

Try to draw the possible lattice structures and you'll get the conclusion.
answered Oct 31, 2019 by neerajpro20 (1,200 points)