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The union of graph G(f) with this other set ?

  1. Jul 23, 2012 #1
    A first year real analysis textbook presents the following two definitions (where the second builds off the first.

    (1) Definition (Graph of a map)

    A and B are sets and [itex]f : A \rightarrow B [/itex] is some map. Then we define the graph of [itex]f[/itex] by [tex]G(f) := \{(x,f(x)) \in A \times B : x \in A\}[/tex].


    (2) Other definition

    A and B are sets and [itex]f : A \rightarrow B [/itex] is some map. Further, define for every [itex]y \in B[/itex] the corresponding intersection [itex]G_{fy}[/itex] by [tex]G_{fy} := G(f) \cap \{(x,y) : x \in A\}[/tex].

    (This then proceeds into a theorem about bijections).

    1. The problem I'm having
    I completely understand (1) and all notation employed in both (1) and (2). However, I don't understand what (2) is trying to communicate ... It seems to me that [itex]G_{fy} = G(f)[/itex] based on my interpretation of (2), making [itex]G_{fy}[/itex] superfluous.
     
  2. jcsd
  3. Jul 23, 2012 #2
    Sorry... I finally figured it out after I posted. [itex]f(x) \in image(f) [/itex], [itex]y \in B[/itex] and image(f) is not necessarily equal to B.

    Feel free to delete.
     
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