1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove a set Identity

Tags:
  1. Aug 8, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that ##A\cap(B\Delta C)=(A\cap B)\Delta(A\cap C)##

    2. Relevant equations


    3. The attempt at a solution

    L.H.S.=##A\cap(B\Delta C)##
    =##A\cap[(B - C) \cup (C - B)]##
    =##A\cap[(B \cap \bar{C}) \cup (C \cap \bar{B})]##
    =##[A\cap (B \cap \bar{C})] \cup [A\cap (C \cap \bar{B})]##
    =##[(A\cap B) \cap \bar{C}] \cup [(A\cap C) \cap \bar{B}]##

    R.H.S.=##(A \cap B) \Delta (A \cap C)##
    =##[(A \cap B) - (A \cap C)] \cup [(A \cap C) - (A \cap B)]##
    =##[(A\cap B) \cap \bar{(A\cap C)} ] \cup [(A\cap C) \cap \bar{(A\cap B)}]##
    =##[(A\cap B) \cap (\bar{A} \cup \bar{C})] \cup [(A\cap C) \cap (\bar{A} \cup \bar{B})]##

    I tried to make L.H.S = R.H.S. But with above results, It's not possible. Can anyone tell me what I've assumed wrong.
     
  2. jcsd
  3. Aug 8, 2015 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Are you required to use that method? The most basic way to prove "X= Y" is to prove both "[itex]X\subseteq Y[/itex]" and "[itex]Y\subseteq X[/itex]"".
    And the most basic way to prove "[itex]X\subseteq Y[/itex]" is to start "if x in in X" and use the definitions and properties of X and Y to conclude "therefore x is in Y".

    Here we want to prove [itex]A\cap\left(B\Delta C\right)= \left(A\cap B\right)\Delta\left(A\cap C\right)[/itex] so we first prove
    [itex]A\cap\left(B\Delta C\right)\subseteq \left(A\cap B\right)\Delta\left(A\cap C\right)[/itex]

    To do that:
    if [itex]x\in A\cap\left(B\Delta C\right)[/itex] then x is in A and x is in B or C but not both. So look at two cases

    1) x is in B but not C. Then x is in [itex]A\cap B[/itex] but not [itex]A\cap C[/itex]. Therefore x is in [itex]\left(A\cap B\right)\Delta\left(A\cap C\right)[/itex].

    2) x is in C but not in B. Then x is in [itex]A\cap C[/itex] but not [itex]A\cap B[/itex]. Therefore x is in [itex]\left(A\cap B\right)\Delta\left(A\cap C\right)[/itex].

    Now show that [itex]\left(A\cap B\right)\Delta\left(A\cap C\right)\subseteq A\cap\left(B\Delta C\right)[/itex] the same way:
    if [itex]x \in \left(A\cap B\right)\Delta\left(A\cap C\right)[/itex] then ...
     
  4. Aug 8, 2015 #3
    I understand your approach. But I want to prove it through Set identities.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Prove a set Identity
  1. Prove Identity (Replies: 6)

  2. Prove the identity. (Replies: 7)

  3. Prove the identity (Replies: 8)

  4. Prove this identity (Replies: 4)

  5. Proving Identities (Replies: 13)

Loading...