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Math Help - Composition relation.

  1. #1
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    Composition relation.

    What is the composition of the relation | (divides) and \le?

    My general interpretation:

    Let R_1 be the | relation, and R_2 be \le. R_1 \circ R_2 should be the set of all points (s, u) where s divides some t \le u.

    If this is correct, (I'm not sure) how would I express this as set-builder notation?

    R_1 \circ R_2 = \lbrace (s, u)\; |\; s | t,\; t \le u \rbrace ?
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  2. #2
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    Quote Originally Posted by scorpion007 View Post
    What is the composition of the relation | (divides) and \le?My general interpretation:
    Let R_1 be the | relation, and R_2 be \le. R_1 \circ R_2 should be the set of all points (s, u) where s divides some t \le u.
    If this is correct, (I'm not sure) how would I express this as set-builder notation?
    R_1 \circ R_2 = \lbrace (s, u)\; |\; s | t,\; t \le u \rbrace ?
    That is very good. Except look at the order of composition. I would add the bit about 'some t' as:
    R_2  \circ R_1  = \left\{ {\left( {s,u} \right):\left( {\exists t} \right)\left[ {s|t\;\& \,t \le u} \right]} \right\}
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  3. #3
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    Ah, I see. Is there a reason why \exists t needs to be parenthesised?

    Thank you.
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  4. #4
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    I'm sorry, why is the order of composition reversed here?
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  5. #5
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    Quote Originally Posted by scorpion007 View Post
    Ah, I see. Is there a reason why \exists t needs to be parenthesised?
    Style
    Be sure to see my edit about order of composition
    It sould be R_2  \circ R_1, remember right to left.
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  6. #6
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    Hmm.. My book states the following regarding a composition:

    Given relations R_1 \subset S \times T and R_2 \subset T \times U, the composition R_1 \circ R_2 consists of all pairs (s, u) \in S \times U for which there is a t \in T with (s, t) \in R_1 and (t, u) \in R_2.
    From that definition, I do not see why the composition of | and \le should be R_2 \circ R_1 given R_1 = | and R_2 = \le as previously.

    Perhaps I missed something?
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  7. #7
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    Think of function composition. Suppose that f(x) = x^2 \;\& \,g(x) = x + 1.
    In the composition f \circ g(x) = f\left( {g(x)} \right) ‘do the g function’ first then we ‘do the f function’:
    we first add one then square the result (x+1)^2.

    On the other hand g \circ f(x) reverses the order: we first square it and add one to the result (x^2+1).

    That also true of relations. The order is from right to left.
    So doing the divisor first and then the less than or equal to, we get R_2  \circ R_1 .
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  8. #8
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    Oh, so R_1 \circ R_2 is "All points (s, u) where s \le t for some t | u? Which is not what we're after. I guess that makes sense, thanks.
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