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Math Help - Relations

  1. #1
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    Relations

    C, D are subsets of A.
    How can i determine if these relations:
    1)C X D union D X C
    2)C X D intersection D X C
    are reflexive symmetric or transitive relations?

    Thanks for any kind of help
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  2. #2
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    Nov 2009
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    Hello rebbeca,

    I will help you with the first one. Reflexivity in this case means (\forall x\in A)[(x,x)\in (C\times D)\cup (D\times C)]. Since C and D are subsets of A, they need not to be equal to A. This leads us to the idea that we may construct an counterexample to the statement of reflexivity.

    Let A=\{1,2,3\}, C=\{1\} and D=\{2\}. Then C\times D=\{(1,2)\} and D\times C=\{(2,1)\}, thus (C\times D)\cup (D\times C)=\{(1,2),(2,1)\}. If we now take an arbitrary element of A, say 1, we see that (1,1) is not an element of (C\times D)\cup (D\times C). This is a rather extreme example, since also (2,2) and (3,3) are not elements of the relation (C\times D)\cup (D\times C).

    Symmetry in this case means (\forall x,y\in A)[(x,y)\in (C\times D)\cup (D\times C) \rightarrow (y,x)\in (C\times D)\cup (D\times C)]. Thus we start by letting x,y be arbitrary elements of A and assume (x,y)\in (C\times D)\cup (D\times C). By the definition of union this means (x,y)\in C\times D or (x,y)\in D\times C. Thus we have to consider two cases.

    1. case: (x,y)\in C\times D

    This means x\in C and y\in D, thus (y,x)\in D\times C. We have therefore (y,x)\in (C\times D)\cup (D\times C).

    I hope my post helps you, so that you are able to do the second case on your own and to solve the problem.

    Best wishes,
    Seppel
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  3. #3
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    Thank you very much!
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