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Thread: Proving Basic Set Operations

  1. #1
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    Proving Basic Set Operations

    Hello,

    Sorry if my questions seem elementary, I am new to real analysis. Anyway I am having trouble with the following:

    1. Prove that the empty set is unique. That is, suppose A and B are empty sets and prove A=B.

    2. Prove, If U=A(union)B and A(intersection)B = empty set, then A=U\B.

    Thank you for your help.

    k
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  2. #2
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    The first one is simple logic.
    The statment that "if $\displaystyle \color{blue}x \in \emptyset$ then $\displaystyle \color{blue}1=2$" is TRUE.
    Recall that a false statement implies any statement and $\displaystyle x \in \emptyset$ is false.
    So both of these are true:
    $\displaystyle x \in \emptyset=A$ then $\displaystyle x \in B$ and $\displaystyle x \in \emptyset=B$ then $\displaystyle x \in A$
    Therefore we have proved $\displaystyle A=B$.
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  3. #3
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    Thank you

    Oh yes, that is so simple that it went right over my head initially. I see exactly what you mean!

    Thank you!
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  4. #4
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    In general, to prove A= B for sets you prove $\displaystyle A\subset B$ and $\displaystyle B\subset A$. And to prove $\displaystyle A\subset B$ you start with "if $\displaystyle x\in A$" and, using properties of A and B, conclude "$\displaystyle x\in B$".

    In this case, if $\displaystyle x\in A$ then $\displaystyle x\in U$= A Union B and x is not in B since A intersect B is empty. Those together say $\displaystyle x\in$ U\B so $\displaystyle A\subset$U\B.

    If $\displaystyle x\in$ U\B then $\displaystyle x\in U$ but x is not in B. Since every member of of U is in either A or B, x must be in A proving U\B$\displaystyle \subset A$.
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  5. #5
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    I see

    Thank you again!

    That makes sense now that I follow your logic. But I am having so much trouble constructing these from scratch! I will keep working at it, maybe it will start coming to me, like magic... ok, that is wishful thinking.

    K
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  6. #6
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    Quote Originally Posted by math-chef View Post
    Hello,

    Sorry if my questions seem elementary, I am new to real analysis. Anyway I am having trouble with the following:

    1. Prove that the empty set is unique. That is, suppose A and B are empty sets and prove A=B.

    2. Prove, If U=A(union)B and A(intersection)B = empty set, then A=U\B.

    Thank you for your help.

    k

    1) Suppose we have two empty sets Φ and Φ',if we can prove Φ=Φ' ,we are done.

    So we know that the empty set is the subset of any other set X,thus:

    ................$\displaystyle \emptyset\subseteq X$ ....for all X...................

    IN this relation put X=Φ' And we get : $\displaystyle \emptyset\subseteq\emptyset'$.................................................. .....1

    Since also Φ' is an empty set we have:

    .................$\displaystyle \emptyset'\subseteq X$ for all X...................................

    And in this relation put X=Φ and we get :$\displaystyle \emptyset'\subseteq\emptyset$.................................................. .......................................2

    Thus by (1) and (2) and equality of sets we get Φ=Φ'.......................
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