Lattice

Question

Answer 1

Answer : A

The set $N$ of all Positive integers, ordered by divisibility, is a Distributive lattice. 

Moreover, The set $D_n$ of all positive integer divisors of a fixed integer $n$, ordered by divisibility, is a distributive lattice.

If You seek a Formal Proof, Here it goes :

We already know that "The set $(D_n,/)$ is a  lattice." (also Can be proved easily)

So, In Order to Prove it Distributive, We need to Prove either one of the Two Distributive rules (Because If One holds then Other also Holds)

So, We need to Prove that $a \vee (b \wedge  c) = (a \vee b) \wedge (a \vee c)$

This Proof holds good for Set of all positive integers as well because the Meet and Join operations are defined in same manner.

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NOTE 1 : $(N,/)$, where $N$ is set of all positive integers, is Not a Bounded lattice as there is No Greatest element.

NOTE 2 : $(N,/)$, where $N$ is set of all Non-negative integers, is a Bounded lattice as there is Greatest element($0$) as well as least element($1$).

NOTE 3 : $(N,/)$, where $N$ is set of all Non-negative integers, is Not a Complemented lattice because except for $0,1$, for any other element, there is No Complement.

NOTE 4 : $(N,/)$, where $N$ is set of all positive integers, is Not a Complemented lattice as it is Not even Bounded lattice.

Comments

NOTE 2 : (N,/), where N is set of all Non-negative integers, is a Bounded lattice as there is Greatest element(0) as well as least element(1).

 can u explain blue shaded part. I am not getting it

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A Lattice is called bounded lattice iff there exists Greatest and least elements.

In $(N,/)$, Where $N$ is set of all Non-negative integers, $1$ is the least element as it divides every other element in $N$. And $0$ is the greatest element as Every other element divides $0$.

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thank u :)