GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
164 views

Consider the lattice D30

a)Draw the Hasse Diagram of D30.

b) Is D30 complemented?

c) Is D30 distributive ?

asked in Set Theory & Algebra by Loyal (3.8k points)  
recategorized by | 164 views

2 Answers

+5 votes
Best answer

(a)

 

(b)In a Complemented Lattice each element has at least one complemented.

(1) 1∨30 =30                  1∧30 = 1         $\Rightarrow$ 1' =30 and 30'=1

(2) 215 = 30                 2∧15 = 1         $\Rightarrow$  2'=15 and 15'=2

(3)310 = 30                  3∧10 = 1         $\Rightarrow$ 3'=10 and 10'=3

(4)56 = 30                    5∧6 = 1            $\Rightarrow$ 5'=6 and 6'=5

Here Every element has one element hence,it is a complemented Lattice.

 

(c)In a Distributive Lattice complement of an element if exists is unique i.e. each element has at most one element.

Here in D30 Every element has unique  complement.Hence, it is Distributive Lattice.  

 

answered by Veteran (29.3k points)  
selected by
very good explanation..
0 votes

D30={1,2,3,5,6,10,15,30}

because D30 is square free so it is boolean algebra and every  boolean algebra is distributed and complemented also

answered by Boss (9.4k points)  
what do you mean by square free ?
Top Users Feb 2017
  1. Arjun

    5502 Points

  2. Bikram

    4280 Points

  3. Habibkhan

    3972 Points

  4. Aboveallplayer

    3076 Points

  5. Debashish Deka

    2646 Points

  6. Smriti012

    2376 Points

  7. sriv_shubham

    2328 Points

  8. Arnabi

    2174 Points

  9. sh!va

    2080 Points

  10. mcjoshi

    1752 Points

Monthly Topper: Rs. 500 gift card

20,960 questions
26,065 answers
59,802 comments
22,237 users