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    By drawing Venn diagrams show that
    A ⊆ B ∩ C if and only if A ∪ B = B and A ∩ C = A.

    How would you do this?
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    (Original post by khanpatel321)
    By drawing Venn diagrams show that
    A ⊆ B ∩ C if and only if A ∪ B = B and A ∩ C = A.

    How would you do this?
    Draw a general Venn diagram with three sets, and identify what it means on that Venn diagram for AuB=B and A int C = A. Then identify what it means on a different copy of the general Venn diagram for A to be a subset of B int C. Compare your results, noting that they are the same.
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    (Original post by Smaug123)
    Draw a general Venn diagram with three sets, and identify what it means on that Venn diagram for AuB=B and A int C = A. Then identify what it means on a different copy of the general Venn diagram for A to be a subset of B int C. Compare your results, noting that they are the same.

    AuB=B I can draw this but don't know how to draw the other one, I shaded the part where they both intersect however this does not include all of A so it can't be equal to A?
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    (Original post by khanpatel321)
    AuB=B I can draw this but don't know how to draw the other one, I shaded the part where they both intersect however this does not include all of A so it can't be equal to A?
    A \cap C = C means that a certain part of the Venn diagram must in fact be empty. It does include all of A, and so the bit which your diagram includes which isn't in C, must in fact not appear on the diagram. I'd denote that by writing in the "A without C" section something like "empty".
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    (Original post by Smaug123)
    A \cap C = C means that a certain part of the Venn diagram must in fact be empty. It does include all of A, and so the bit which your diagram includes which isn't in C, must in fact not appear on the diagram. I'd denote that by writing in the "A without C" section something like "empty".
    do you mean = A ? not C?
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    (Original post by khanpatel321)
    do you mean = A ? not C?
    Yes, sorry.
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    The mark scheme say's this

    Solution:
    An obvious drawing: in the Venn diagram for A∪B,
    the bit of A which does not belong to B sticks out of B; in the

    Venn diagram for A ∩ C, the bit of A which does not belong
    to C sticks out of A ∩ C




    What do you do next?
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    bump
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    (Original post by khanpatel321)
    bump
    Better to quote me directly like this, so that I get a notification

    I've re-phrased what you've written, and made one further step: drawing the Venn diagram corresponding to "A union B = B, and A int C = A". Draw the Venn diagram corresponding to "A subset (B intersect C)"; you should find that you've ended up with the same diagram.

    Incidentally, this question is much easier without Venn diagrams. A \cap C = A implies "everything in A is also in C" (because everything in the right-hand side must be in the left-hand side, and therefore must be in both A and C). That is, A is a subset of C. And so on.
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    (Original post by Smaug123)
    Better to quote me directly like this, so that I get a notification

    I've re-phrased what you've written, and made one further step: drawing the Venn diagram corresponding to "A union B = B, and A int C = A". Draw the Venn diagram corresponding to "A subset (B intersect C)"; you should find that you've ended up with the same diagram.

    Incidentally, this question is much easier without Venn diagrams. A \cap C = A implies "everything in A is also in C" (because everything in the right-hand side must be in the left-hand side, and therefore must be in both A and C). That is, A is a subset of C. And so on.
    Ohh I see, thanks
 
 
 
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