Results 1 to 2 of 2

Math Help - Inquiry about proofs involving families of sets

  1. #1
    Newbie
    Joined
    Aug 2011
    Posts
    18

    Question Inquiry about proofs involving families of sets

    This post does not concern a particular problem or exercise, but instead a peculiarity (for me) in one genre: proofs involving families of sets (that is, sets containing sets as elements). I have noticed that in some statements of theorems which involve families of sets, the hypothesis includes " let F and G be families of sets," whereas in others, the hypothesis is slightly altered to: "let F and G be nonempty families of sets." I have included two theorems (which I have already proven) as examples of this:

    1. Suppose F and G are nonempty families of sets. Prove that U(F U G) = (UF) U (UG).

    2. Suppose F and G are families of sets. Prove that U(F ∩ G) ⊆ (UF) ∩ (UG).


    The difference obviously regards some property about the empty set. My original thought was that being nonempty allowed for the assertion of F or G containing some set- which arises during the course of the proof of 1. That is, x ∈ U(F U G) means there is some A ∈ F U G for which x ∈ A. But then, however, i noticed that the proof of theorem 2. also asserts the existence of some set which is an element of F (and G), without the "nonempty" portion of the hypothesis being present. Could this simply be a mistake on the part of the author, or am i missing a notion here?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,540
    Thanks
    780

    Re: Inquiry about proofs involving families of sets

    By definition, x ∈ ⋃F iff there exists an A ∈ F such that x ∈ A. Therefore, ⋃∅ = ∅. So, both claims above are true regardless whether F and G are empty. On the other hand, x ∈ ⋂F iff for all A ∈ F it is the case that x ∈ A. If there is an assumed universal set U, then ⋂∅ = U. However, in general ⋂∅ is not defined because there is no set of all sets. Thus, the claim ⋂(F ∪ G) ⊆ ⋂F ∪ ⋂G makes sense only for nonempty F and G.

    One guess why the first statement in the OP requires nonempty sets is that the author knew that either general intersections or general unions are not defined for empty families but did not want to remember which one is not defined.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] problem involving families of sets
    Posted in the Discrete Math Forum
    Replies: 17
    Last Post: July 25th 2011, 06:32 AM
  2. Proofs involving logs
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 26th 2010, 05:36 AM
  3. Help with two proofs involving transversals
    Posted in the Geometry Forum
    Replies: 2
    Last Post: February 17th 2010, 08:56 AM
  4. Try these proofs involving sets
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: April 15th 2009, 02:29 AM
  5. Proofs involving sets and their compliments
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 14th 2009, 11:06 PM

Search Tags


/mathhelpforum @mathhelpforum