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Math Help - Can someone help me with a question about functions?

  1. #1
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    Can someone help me with a question about functions?

    Let f be a function on A onto B. Define a relation E in A by: aEb if and only if f(a)=f(b).

    a. Show that E is an equivalence relation on A.
    b. Define a function phi on A/E onto B by phi([a]_E)=f(a) (verify that phi([a]_E) = phi([a']_E) if [a]_E = [a']_E)
    c. Let j be the function on A onto A/E given by j(a)=[a]_E. Show that phi (dot) j = f.

    _ is sub.
    Any ideas would be really appreciated! Thanks!
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  2. #2
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    Quote Originally Posted by hammertime84 View Post
    Let f be a function on A onto B. Define a relation E in A by: aEb if and only if f(a)=f(b).

    a. Show that E is an equivalence relation on A.
    b. Define a function phi on A/E onto B by phi([a]_E)=f(a) (verify that phi([a]_E) = phi([a']_E) if [a]_E = [a']_E)
    c. Let j be the function on A onto A/E given by j(a)=[a]_E. Show that phi (dot) j = f.

    _ is sub.
    Any ideas would be really appreciated! Thanks!
    Well, for (a) use the definition of "equivalence relation"!

    a) (reflexive). For every x in A xEx. If x in A is f(x)= f(x)?

    b) (symmetric). For all x, y in A if xEy then yEx. If x and y are in A and f(x)= f(y) is f(y)= f(x)?

    c) (transitive). For all x, y, z, in A, if xEy and yEz then xEz. If x, y, z are in A, f(x)= f(y) and f(y)= f(z) is f(x)= f(z)?

    b) "A/E" is the set consisting of all "equivalence classes"- subsets of A such that every x, y in that subset have xEy.
    Did this problem actually say "if [a]_E = [a']_E"? In that case this is trivial. By definition of "function" if f is a function and u= v then f(u)= f(v). Much more interesting would be "verify that phi([a]_E) = phi([a']_E) if a'Ea ". But that's still easy. If a'Ea then what can you say about f(a') and f(a)?

    c) Not "phi dot j", phi \circ j, the composition. Okay, for any a, j(a) is the equivalence class containing a. What is true about f(x) for every member, x, of that class?
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  3. #3
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    Thanks for your help, but I'm still kind of confused about c. Could you elaborate a bit?
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