Results 1 to 3 of 3

Math Help - de Morgan's laws on generalised union and intersection.

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    62

    de Morgan's laws on generalised union and intersection.

    Hi, I need to prove de Morgan's laws for generalised union and intersections, I came up with a proof but I don't know if it's works with what I know.
    Here is what I know:
    Given A, a set and B such that B \subseteq A then  B^{c}=A-B.
    \mathcal{A} is a collection of sets.
     \bigcup_{A \in \mathcal{A}}A=\{ x| \exits A \in \mathcal {A} \text{ such that } x \in A \}.
     \bigcap_{A \in \mathcal{A}}A= \{x| x \in A, \forall A \in \mathcal A\}.

    proof:
     \text{1. }(\bigcup_{A \in \mathcal{A}}A)^{c}=\{x| x \notin A, \forall A \in \mathcal{A}\}=\{x|x \in A^{c}, \forall A \in \mathcal{A}\}=\bigcap_{A \in \mathcal{A}}A^{c}.
    \text{2. }(\bigcap_{A \in \mathcal{A}}A)^{c}=\{x| \exits A \in \mathcal{A} \text{ such that } x \notin A\}= \{ x| \exists A \in \mathcal{A} \text{ such that } x \in A^{c}\}=\bigcup_{A \in \mathcal{A}}A^c.

    Thanks in advance!.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,502
    Thanks
    765

    Re: de Morgan's laws on generalised union and intersection.

    I think the proof is OK.

    I wanted to make a remark about the lack of a quantifier over A in the definition of union, but then I saw that it is a LaTeX error. Since you put existential quantifier in the beginning, I would do the same with the universal quantifier. I would also add an extra equality \left(\bigcup_{A \in \mathcal{A}}A\right)^{c}=\{x\mid\neg\exists A\in\mathcal{A}.\,x\in A\} for clarity, and similarly for intersection.

    Hint: use \mid instead of | in the set-builder notation. It creates correct spaces around it.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2009
    Posts
    62

    Re: de Morgan's laws on generalised union and intersection.

    Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. De Morgan's Laws proof
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 17th 2011, 02:56 AM
  2. 2 Questions about probability+De Morgan laws
    Posted in the Statistics Forum
    Replies: 11
    Last Post: April 12th 2011, 07:20 PM
  3. proof of De Morgan laws
    Posted in the Math Challenge Problems Forum
    Replies: 5
    Last Post: June 20th 2010, 05:02 AM
  4. De Morgan's laws
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: December 18th 2008, 01:16 AM
  5. de morgan's laws
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: October 24th 2008, 05:17 PM

Search Tags


/mathhelpforum @mathhelpforum