GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
263 views
In one text I read that , if n is square free it is DISTRIBUTIVE
in other text I read that if n is square free  it is BOOLEAN ALGEBRA .

Which is most correct ?

Here D36 is not square free then... what conclusion can I make ?
asked in Set Theory & Algebra by Veteran (20.7k points)   | 263 views

We talk about distributivity and boolean algebra only iff it's a lattice(POSET). So, Relation needs to be provided with Set under discussion.

So, the correct question would go like : let $[D_{36},/]$ (or) Set $D_{36}$ on relation divides. (although its know that we talk about division mostly)

5 Answers

+1 vote
Best answer
You can simply check whether d36 is distributive  or not by checking whether we can get it by product of distinct primes for example
D30=2.3.5=30 possible  d40=2.2.2.5 not possible as 2 came 3 times not distinct d36=2.3.2.3 not distributive as we did not get it through product of distinct primes
answered by Ajay singh  
selected ago by
+1 vote
Dn if n is a square free number,then  it will be a boolean algebra because if it is perfect square ,then its factors will repeat and then it may lead to more than one complement of an element which is actually not a boolean algebra. ...so D36 is not a boolean algebra.
answered by Boss (8.7k points)  
How do you account for distributive ?
+1 vote
When we draw the  hasse diagram, 6 doesnt have a complement. Now, for a lattice to be distributive, every element must have unique complement.

Hence, not distributive.
answered by Veteran (15.2k points)  
+1 vote

In Dn ,if n is square free number then it will be a boolean algebra along with the numbe of vertex should be 2^n and number of edges should be 2*2^n-2.

Above is most important condition to identify whether a relation is boolean algebra or not.Above rule is not for distributive lattice.

Distributive lattice fallow the distributive properties and sublattice properties.

Example:: D64 not Boolean algebra but D110 is boolean algebra.

answered by Loyal (4.8k points)  
0 votes

D36 is not a boolean algebra and not a distributive lattice .

 
36 = 3*2*2*3 so 2 and 3 is repeating hence it is not distributive .

We did not get 36 as product of distinct primes , hence it is not distributive .

answered ago by Veteran (33k points)  


Top Users Jul 2017
  1. Bikram

    4910 Points

  2. manu00x

    2940 Points

  3. Debashish Deka

    1870 Points

  4. joshi_nitish

    1776 Points

  5. Arjun

    1506 Points

  6. Hemant Parihar

    1306 Points

  7. Shubhanshu

    1128 Points

  8. pawan kumarln

    1124 Points

  9. Arnab Bhadra

    1114 Points

  10. Ahwan

    956 Points


24,099 questions
31,074 answers
70,703 comments
29,407 users