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Thread: Proving sets equality

  1. #1
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    Proving sets equality

    A and B are sets,
    Prove that if 𝐴∖𝐵=∅ and 𝐵∖𝐴=∅ then 𝐴=𝐵 .

    Now i know it has something to do saying that X is in set A or something like that but im kinda stuck.
    Any help? need the general idea

    also have a thing where i dont know how to actually write the solution in this :

    A,B,C are sets,
    prove that if 𝐴∩𝐶⊆𝐵∩𝐶 and 𝐴∩𝐶̅'⊆𝐵∩𝐶' then 𝐴⊆𝐵 (using ' for NOT)

    I know that 𝐴∩𝐶 and 𝐴∩𝐶̅' will give me everything in A, and the same for the other side will give me B but i dont know how to write it in an actual solution form.
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  2. #2
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    Re: Proving sets equality

    Quote Originally Posted by Wesix View Post
    A and B are sets,
    Prove that if 𝐴∖𝐵=∅ and 𝐵∖𝐴=∅ then 𝐴=𝐵 .
     \begin{align*}A\setminus B&=A\cap B^c=\emptyset \\B\setminus A&=B\cap A^c=\emptyset\\ [A\cap B^c=\emptyset] & \Rightarrow A\subseteq B\\ [B\cap A^c=\emptyset] & \Rightarrow B\subseteq A\end{align*}

    Can you finish?
    Thanks from Wesix and topsquark
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    Re: Proving sets equality

    Quote Originally Posted by Plato View Post
     \begin{align*}A\setminus B&=A\cap B^c=\emptyset \\B\setminus A&=B\cap A^c=\emptyset\\ [A\cap B^c=\emptyset] & \Rightarrow A\subseteq B\\ [B\cap A^c=\emptyset] & \Rightarrow B\subseteq A\end{align*}

    Can you finish?
    Yes that helped, Ty.
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