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Math Help - How do I solve this with a direct proof?

  1. #1
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    How do I solve this with a direct proof?

    Prove that for all sets X and Y, (X ∩ Y) ∪(Y-X)=Y

    I'm stuck and I'm not fully sure how to solve this.
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  2. #2
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    Re: How do I solve this with a direct proof?

    suppose that a is an element of (X∩Y) U (Y-X).

    this means one of two things:

    a is in X∩Y, or
    a is in Y-X

    suppose a is in X∩Y. this means that a is in BOTH X and Y, so is certainly in Y.

    alternatively, if a is in Y-X, this means a is in Y, but not in X. who cares if its not in X, at least it's in Y!

    so in all possible cases, we see that a is in Y.

    this means that (X∩Y) U (Y-X) is a subset of Y.

    now suppose b is in Y.

    well we have two possibilities:

    b is in X
    b is NOT in X (in or out, that's the way it is with sets. you cannot be "sort of" in a set).

    if b is in X, then b is in X AND Y, so b is in X∩Y.
    if b is not in X, then b is in Y, but not in X, so b is in Y-X.

    if we put these two sets together, b is certain to be in one of them. hence b is in (X∩Y) U (Y-X).

    thus Y is a subset of (X∩Y) U (Y-X).

    but, if for 2 sets A,B: if A⊆B and B⊆A, then A and B have exactly the same elements, that is: A = B.

    so (X∩Y) U (Y-X) = Y.

    (intuitively what we are doing is splitting Y into 2 parts: the part that overlaps with X, and the part that doesn't).
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    Re: How do I solve this with a direct proof?

    Hello, blueaura94!

    \text{Prove: }\:(X \cap Y) \cup (Y - X) \:=\:Y

    \begin{array}{ccccc}1. & (X \cap Y) \cup (Y - X) && 1. & \text{Given} \\ 2. & (X \cap Y) \cup (Y \cap \overline{X}) && 2. & \text{Def. Subtr'n} \\ 3. & (X\cap Y) \cup (\overline{X} \cap Y) && 3. & \text{Commutative} \\ 4. & (X \cup \overline{X}) \cap Y && 4. & \text{Distributive} \\ 5. & U \cap Y && 5. & A \cup \overline{A} \,=\,\mathbb{U} \\ 6. & Y && 6. &  \mathbb{U} \cap A \,=\,A \end{array}
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    Re: How do I solve this with a direct proof?

    Quote Originally Posted by blueaura94 View Post
    Prove that for all sets X and Y, (X ∩ Y) ∪(Y-X)=Y
    Here is another way.
    X\cap Y\subseteq Y~\&~Y\setminus X\subseteq Y so there is just need to prove one way.

    If t\in Y then either t\in X or t\notin X.

    And we are done.
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