Results 1 to 14 of 14

Math Help - Boolean Algebra

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Boolean Algebra

    In order to show a Lattice is a Boolean Algebra, the diagram needs to be bounded, distributive, complemented, and |L|=2^n \ n\geq 1 \ n\in\mathbb{Z}.

    However, my book says, "A finite lattice is called a Boolean Algebra if it is isomorphic to B_n for some nonnegative integer n."

    What is B_n?

    I know D_n are the numbers that divide n but have no clue about this B_n.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    In order to show a Lattice is a Boolean Algebra, the diagram needs to be bounded, distributive, complemented, and |L|=2^n \ n\geq 1 \ n\in\mathbb{Z}.

    However, my book says, "A finite lattice is called a Boolean Algebra if it is isomorphic to B_n for some nonnegative integer n."

    What is B_n?

    I know D_n are the numbers that divide n but have no clue about this B_n.

    Thanks.
    I'm not quite sure what they mean, but a fundamental result in the study of Boolean algebras is that every finite Boolean algebra B is isomorphic to 2^{[n]} (here [n]=\{1,\cdots,n\}) for some n\in\mathbb{N}. This fact would make me guess that B_n=2^{[n]}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    So if \displaystyle |L|=8, what am I showing it is isomorphic too?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    So if \displaystyle |L|=8, what am I showing it is isomorphic too?
    2^{[3]}.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by Drexel28 View Post
    2^{[3]}.
    Unfortunately, that didn't help. I understand 8=2^3 but how do I show it is isomorphic to  2^3?
    Last edited by dwsmith; December 7th 2010 at 04:50 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    Unfortunately, that didn't help. I understand 8=2^3 but how do I show is isomorphic to  2^3?
    This isn't in general easy. The result takes a fair amount of background. Do you know what 'atomic' means?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Haven't a clue unless you are talking about the science sense or as in bomb.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    Haven't a clue unless you are talking about the science sense or as in bomb.
    Hahaha. Nice. Then what do you have?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    The book mentions making a Hasse diagram of B_n. What would that look like?

    Here is the Hasse diagram I need to show it is isomorphic to B_n.

    Labeled a to h. a=O and h=I

    Boolean Algebra-hasse-diagram.jpg
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    The book mentions making a Hasse diagram of B_n. What would that look like?

    Here is the Hasse diagram I need to show it is isomorphic to B_n.

    Labeled a to h. a=O and h=I

    Click image for larger version. 

Name:	Hasse Diagram.jpg 
Views:	4 
Size:	10.9 KB 
ID:	20015
    Here is the Hasse diagram for 2^{[3]}
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    How is that the Hasse Diagram for 2^3\mbox{?} How do I come up with the Hasse diagram when I have 2^n is the better question?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    How is that the Hasse Diagram for 2^3\mbox{?} How do I come up with the Hasse diagram when I have 2^n is the better question?
    Because it's the set of all subsets of a set with three elements. You do the same thing. Just draw out the diagram of the lattice for the power set of \{1,2\}.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    So 4 is the P(s)={1,2,3,4} for instance?
    Follow Math Help Forum on Facebook and Google+

  14. #14
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by dwsmith View Post
    So 4 is the P(s)={1,2,3,4} for instance?
    What I mean, is that if you have a Boolean algebra B with \#(B)=2^n then B\cong\mathcal{P}\left(\{1,\cdots,n\}\right) if that notation is more comfortable.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] boolean algebra help
    Posted in the Statistics Forum
    Replies: 1
    Last Post: October 10th 2011, 03:06 PM
  2. Boolean algebra Help
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: January 11th 2010, 02:15 PM
  3. Boolean algebra help
    Posted in the Math Topics Forum
    Replies: 6
    Last Post: November 21st 2009, 02:50 PM
  4. boolean algebra
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: January 22nd 2009, 09:57 AM
  5. Boolean Algebra
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: February 8th 2008, 08:55 AM

/mathhelpforum @mathhelpforum