Results 1 to 2 of 2

Math Help - How do you distribute AND and OR?

  1. #1
    Junior Member
    Joined
    Mar 2008
    Posts
    40

    How do you distribute AND and OR?

    Hi all,
    I'm working through a book on set theory and I haven't gotten very far yet. I'm sort of struck by the looseness of the proofs for the associative, distributive, and commutative laws provided by the book and I think this stems from the fact that I'm not sure how to relate / distribute the ANDs and ORs of written logic.

    Example:
    If x is an element of (A intersect B) union C, then x is an element of A AND x is an element of B, OR x is an element of C.

    Why does the proof jump from this kind of formulation to: (A U C) intersect (B U C)?
    Why is the OR back distributed over the AND?

    Any help clarifying my confusion would be much appreciated.
    Thanks,
    Ultros
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Smile

    Quote Originally Posted by Ultros88 View Post
    Hi all,
    I'm working through a book on set theory and I haven't gotten very far yet. I'm sort of struck by the looseness of the proofs for the associative, distributive, and commutative laws provided by the book and I think this stems from the fact that I'm not sure how to relate / distribute the ANDs and ORs of written logic.

    Example:
    If x is an element of (A intersect B) union C, then x is an element of A AND x is an element of B, OR x is an element of C.

    Why does the proof jump from this kind of formulation to: (A U C) intersect (B U C)?
    Why is the OR back distributed over the AND?

    Any help clarifying my confusion would be much appreciated.
    Thanks,
    Ultros
    Using \wedge for AND and \vee for OR, we have the following properties:

    (A\wedge B)\vee C\Leftrightarrow (A\vee C)\wedge(B\vee C)

    (A\vee B)\wedge C\Leftrightarrow (A\wedge C)\vee (B\wedge C)

    \emph{Proof: } We use a truth table (note that \text{T} means "true" and \text{F} "false"):

    \begin{tabular}{c|c|c|c|c}<br />
 $A$ & $B$ & $C$ & $A\wedge B$ & $(A\wedge B)\vee C$\\\hline<br />
 T & T & T &       T       & T\\<br />
 T & T & F &        T       & T\\<br />
 T & F & T &       F       & T\\<br />
 T & F & F &       F       & F\\<br />
 F & T & T &       F       & T\\<br />
 F & T & F &       F       & F\\<br />
F & F & T &       F       & T\\<br />
F & F & F &       F       & F<br />
\end{tabular}

    \begin{tabular}{c|c|c|c|c|c}<br />
 $A$ & $B$ & $C$ & $A\vee C$ & $B\vee C$ & $(A\vee C)\wedge (B\vee C)$\\\hline<br />
 T & T & T &       T       & T & T\\<br />
 T & T & F &        T       & T & T\\<br />
 T & F & T & T       & T & T\\<br />
 T & F & F & T       & F & F\\<br />
 F & T & T & T       & T & T\\<br />
 F & T & F & F       & T & F\\<br />
F & F & T & T       & T & T\\<br />
F & F & F &       F       & F & F<br />
\end{tabular}

    Since the final columns of the two truth tables are the same, we can say that the two statements are equivalent.

    Using a similar method, you can show that (A\vee B)\wedge C\Leftrightarrow (A\wedge C)\vee (B\wedge C) \square


    Thus, we can show that x\in(A\cap B)\cup C\Leftrightarrow x\in(A\cup C)\cap(B\cup C)

    \emph{Proof: }

    x\in(A\cap B)\cup C\Leftrightarrow (x\in A\cap B)\vee(x\in C)

    \Leftrightarrow (x\in A\wedge x\in B)\vee(x\in C)

    \Leftrightarrow (x\in A\vee x\in C)\wedge(x\in B\vee x\in C) (from the result above)

    \Leftrightarrow (x\in A\cup C)\wedge(x\in B\cup C)

    \Leftrightarrow x\in(A\cup C)\cap(B\cup C)\quad\square
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Distribute subjects' ages for most Power
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: September 22nd 2011, 04:13 PM
  2. Replies: 0
    Last Post: March 15th 2011, 05:54 AM
  3. Distribute, Expect!
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: August 5th 2010, 04:59 AM
  4. Distribute 71.12
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 28th 2009, 10:32 AM
  5. Distribute and Simplify help
    Posted in the Algebra Forum
    Replies: 8
    Last Post: April 27th 2009, 06:09 AM

Search Tags


/mathhelpforum @mathhelpforum