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Math Help - Can someone help me with my proof?

  1. #1
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    Can someone help me with my proof?

    Hello,
    I have been working on this proof for quite some time now can someone give me some pointers?

    Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.


    I know that if somehow I can prove that A intersected with B= empty set then I can use the laws of transitivity to come to prove my proof. I also know that if B intersected with C = empty set that means both B AND C are empty and contain no elements. If 'all the members' of A are in B and B= empty set then A= empty set as well.



    Can someone help me piece this together please?
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  2. #2
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    Suppose that x\in A\cap C.
    That means that x\in A. But that means x\in B.
    What is wrong with that?
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  3. #3
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    I am not sure, it is hard for me to come up with an answer from that hint thanks though....


    I was thinking something along the lines of "x cannot be both an element of A and an element of B because......." but then I get stuck.
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  4. #4
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    Quote Originally Posted by matthayzon89 View Post
    Hello,
    I have been working on this proof for quite some time now can someone give me some pointers?

    Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.


    I know that if somehow I can prove that A intersected with B= empty set then I can use the laws of transitivity to come to prove my proof. I also know that if B intersected with C = empty set that means both B AND C are empty and contain no elements. If 'all the members' of A are in B and B= empty set then A= empty set as well.



    Can someone help me piece this together please?

    you want to prove that :

    A\cap C =\emptyset

    Because you know that the empty set is a subset of each set ,then all you have to prove is :


    A\cap C\subset\emptyset

    OR

    xεΑ and xεC => x\in\emptyset

    So start with : xεΑ AND xεC and see where this will lead you to
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  5. #5
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    Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.

    Okay so this is my proof attempt:

    Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

    We know that empty set is the subset of each set. So, empty set is the subset A, empty set is the subset of B and empty set is the subset of C.

    Since, empty set is the the subset of A and B then A intersect B is an empty set. If A intersect B= empty set and B intersect C equals empty set then A intersect C equals empty set (By laws of transitivity). **END OF PROOF**



    P.S
    Is the set A with a bar on top of it notation the same as the complement of A?
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  6. #6
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    Quote Originally Posted by matthayzon89 View Post
    Prove: If A is a subset of B and B intersected with C= empty set then A intersected with C= empty set.

    Okay so this is my proof attempt:

    .

    Since, empty set is the the subset of A and B then A intersect B is an empty set. If
    WRONG :

    Let A ={1,2} B={1,5} IS A\cap B=\emptyset????


    Yet \emptyset\subset A\cap B
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  7. #7
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    I GET THE GENERAL IDEA NOW!

    Since A is the subset of B (A is IN B) and B intersect C have no elements in common (empty set, meaning they don't 'overlap'), then OF COURSE that A and C DONT have any elements in common as well (they also dont overlap because A is IN B).
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  8. #8
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    Quote Originally Posted by matthayzon89 View Post
    lol yup, your right.

    Darn this problem! can someone give me another hint please? ...im as lost as can be.

    Ι told you to start with xεΑ and xεC.

    Since A\subset B => xεA => xεB.

    You also have xεC ,SO now you have that xεB and xεC .

    BUT .....can you curry on from here now??
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  9. #9
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    Quote Originally Posted by matthayzon89 View Post
    I GET THE GENERAL IDEA NOW!

    Since A is the subset of B (A is IN B) and B intersect C have no elements in common (empty set, meaning they don't 'overlap'), then OF COURSE that A and C DONT have any elements in common as well (they also dont overlap because A is IN B).
    Yes ,but you have to show that
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  10. #10
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    Quote Originally Posted by xalk View Post
    Yes ,but you have to show that
    Here it is:

    Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

    Let x be an element of set A. If x E A and B intersect C= empty set, then x is NOT an element of C. Which means A is NOT a subset of C. Therefore, A intersect C= empty set as well (A and C cannot intersect and result in a NON-empty set if A is not a subset of C). If A intersect C= empty set and B intersect C= empty set, then A intersect C= EMPTY SET by laws of transitivity.
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  11. #11
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    Quote Originally Posted by matthayzon89 View Post
    Here it is:

    Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

    Let x be an element of set A. If x E A and B intersect C= empty set, then x is NOT an element of C.
    Let:

    A={1,2}, B={3,5} ,C={1,7} .Here we have 1εA , B\cap C=\emptyset .
    Yet 1εC
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  12. #12
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    Can someone please help?! This is frustrating

    How about now...

    Suppose that A, B, and C are sets where A is a subset of B and B intersect C= empty set.

    Let x be an element of set A. If x E A and B intersect C= empty set, then A is NOT a subset of C. Therefore, A intersect C= empty set as well (A and C cannot intersect and result in a NON-empty set if A is not a subset of C). If A intersect C= empty set and B intersect C= empty set, then A intersect C= EMPTY SET by laws of transitivity.
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  13. #13
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    Quote Originally Posted by matthayzon89 View Post
    Can someone please help?! This is frustrating
    Cool down ,please read my posts again ,get out of your mind the transitivity law

    When you solve a problem ,think before write or say anything.

    Ask yourself what is given in a problem and what is asked to prove .


    In a proof you must use all the given ,otherwise some where you must have gone wrong
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  14. #14
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    Quote Originally Posted by xalk
    In a proof you must use all the given ,otherwise some where you must have gone wrong
    My friend forgot to do that in a course final exam, but apparently he only proved a stronger result, so it's not always bad :P

    On topic, matthayzon89 - take xalk's advice: Let x \in A \cap C, then x \in A and x \in C. Using the information you are given, what will you do next?
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  15. #15
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    Suppose that A,B, and C are sets.
    Let x E A and x E C.
    A is a subset of B. If A is a subset of B and x is an element in A then x is ALSO an element in B.
    If the intersect of all the elements of B AND C= empty set
    we can conclude that x= empty set?
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