# 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?

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

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

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

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

reshown Jul 9, 2019

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 (34,770 points)
{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
@tsutkarsh
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
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